What are the reasons to expect that gravity should be quantized? What I am interested to see are specific examples/reasons why gravity should be quantized. Something more than "well, everything else is, so why not gravity too". For example, isn't it possible that a quantum field theory on curved space-time would be the way treat QFT and gravity in questions where the effects of neither can be ignored?
 A: For the sake of argument, I might offer up a plausible alternative.  We might have some quantum underpinning to gravitation, but we might in fact not really have quantum gravity.  It is possible that gravitation is an emergent phenomenon from a quantum field theoretic substratum, where the continuity of spacetime might be similar to the large scale observation of superconductivity or superfluidity.  The AdS/CFT is a matter of classical geometry and its relationship to a quantum field theory.  So the $AdS_4/QFT$ suggests a continuity of spacetime which has a correspondence with the quark-gluon plasma, which has a Bjorken hydrodynamic scaling.  The fluid dynamics of QCD, currently apparent in some LHC and RHIC heavy ion physics, might hint at this sort of connection.
So we might not really have a quantum gravity as such. or if there are quantum spacetime effects it might be more in the way of quantum corrections to fluctuations with some underlying quantum field.  Currently there are models which give quantum gravity up to 7 loop corrections, or 8 orders of quantization.  Of course the tree level of quantum gravity is formally the same as classical gravity. 
This is suggested not as some theory I am offering up, but as a possible way to think about things.
A: Gravity has to be subject to quantum mechanics because everything else is quantum, too. The question seems to prohibit this answer but that can't change the fact that it is the only correct answer. This proposition is no vague speculation but a logically indisputable proof of the quantumness.
Consider a simple thought experiment. Install a detector of a decaying nucleus, connected to a Schrödinger cat. The cat is connected to a bomb that divides the Earth into two rocks when it explodes. The gravitational field of the two half-Earths differs from the gravitational field of the single planet we know and love.
The nucleus is evolving into a superposition of several states, inevitably doing the same thing with the cat and with the Earth, too. Consequently, the value of the gravitational field of the place previously occupied by the Earth will also be found in a superposition of several states corresponding to several values - because there is some probability amplitude for the Earth to have exploded and some probability amplitude for it to have survived.
If it were possible to "objectively" say whether the gravitational field is that of one Earth or two half-Earths, it would also be possible to "objectively" say whether the nucleus has decayed or not. More generally, one could make "objective" or classical statements about any quantum system, so the microscopic systems would have to follow the logic of classical physics, too. Clearly, they don't, so it must be impossible for the gravitational field to be "just classical".
This is just an explicit proof. However, one may present thousands of related inconsistencies that would follow from any attempt to combine quantum objects with the classical ones in a single theory. Such a combination is simply logically impossible - it is mathematically inconsistent.
In particular, it would be impossible for the "classical objects" in the hybrid theory to evolve according to expectation values of some quantum operators. If this were the case, the "collapse of the wave function" would become a physical process - because it changes the expectation values, and that would be reflected in the classical quantities describing the classical sector of the would-be world (e.g. if the gravitational field depended on expectation values of the energy density only). 
Such a physicality of the collapse would lead to violations of locality, Lorentz invariance, and therefore causality as well. One could superluminally transmit the information about the collapse of a wave function, and so on. It is totally essential for the consistency of quantum mechanics - and its compatibility with relativity - to keep the "collapse" of a wave function as an unphysical process. That prohibits observable quantities to depend on expectation values of others. In particular, it prohibits classical dynamical observables mutually interacting with quantum observables.
A: I have seen two converging paths as compelling reasons for quantizing gravity, both dependent on experimental observations.
One is the success of gauge theories in particle physics the past decades, theories that organized knowledge mathematically economically and elegantly. Gravitational equations are very tempting since they look like a gauge theory.
The other is the Big Bang theory of the beginning of the universe that perforce has to evolve  the generation of particles and interactions from a unified model, as the microseconds grow. It is attractive and elegant that the whole is unified in a quantum theory that evolves into all the known interactions, including gravity.
A: Reasons for why gravity should be amenable to "quantization":


*

*Because everything else or as @Marek puts it because "the world is inherently quantum". This in itself is more an article of faith than an argument per se.

*Because QFT on curved spacetime (in its traditional avatar) is only valid as long as backreaction is neglected. In other words if you have a field theory then this contributes to $T_{\mu\nu}$ and by Einstein's equations this must in turn affect the background via:
$$ G_{\mu\nu} = 8\pi G T_{\mu\nu} $$
Consequently the QFTonCS approach is valid only as long as we consider field strengths which do not appreciable affect the background. As such there is no technical handle on how to incorporate backreaction for arbitrary matter distributions. For instance Hawking's calculation for BH radiation breaks down for matter densities $\gt M_{planck}$ per unit volume and possibly much sooner. Keep in mind that $M_{planck}$ is not some astronomical number but is $\sim 21 \, \mu g$, i.e. about the mass of a colony of bacteria!
The vast majority of astrophysical processes occur in strong gravitational fields with high enough densities of matter for us to distrust such semiclassical calculations in those regimes.

*Well there isn't really a good third reason I can think of, other than "it gives you something to put on a grant proposal" ;)
So the justification for why boils down to a). because it is mandatory and/or would be mathematically elegant and satisfying, and b). because our other methods fail in the interesting regimes.
In the face of the "inherently quantum" nature of the world we need strong arguments for why not. Here are a couple:


*

*The world is not only "inherently quantum" but it is also "inherently geometric" as embodied by the equivalence principle. We know of no proper formulation of QM which can naturally incorporate the background independence at the core of GR. Or at least this was the case before LQG was developed. But LQG's detractors claim that in the absence of satisfactory resolutions of some foundational questions (see a recent paper by Alexandrov and Roche, Critical overview of Loops and Foams). Also despite recent successes it remains unknown as to how to incorporate matter into this picture. It would appear that topological preons are the most natural candidates for matter given the geometric structure of LQG. But there does not appear to be any simple way of obtaining these braided states without stepping out of the normal LQG framework. A valiant attempt is made in this paper but it remains to be seen if this line of thought will bear sweet, delicious fruit and not worm-ridden garbage!

*Starting with Jacobson (AFAIK) (Thermodynamics of Spacetime: The Einstein Equation of State, PRL, 1995) there exists the demonstration that Einstein's equations arise naturally once one imposes the laws of thermodynamics ($dQ = TdS$) on the radiation emitted by the local Rindler horizons as experienced by any accelerated observer. This proof seems to suggest that the physics of horizons is more fundamental than Einstein's equations, which can be seen as an equation of state. This is analogous to saying that one can derive the ideal gas law from the assumption that an ideal gas should satisfy the first and second laws of thermodynamics in a suitable thermodynamical limit ($N, V \rightarrow \infty$, $N/V \rightarrow$ constant). And the final reason for why not ...

*Because the other, direct approaches to "quantizing" gravity appear to have failed or at best reached a stalemate.
On balance, it would seem that one can find more compelling reasons for why not to quantize gravity than for why we should do so. Whereas there is no stand-alone justification for why (apart from the null results that I mention above), the reasons for why not have only begun to multiply. I mention Jacobson's work but that was only the beginning. Work by Jacobson's student (?) Christopher Eling (refs) along with Jacobson and a few others has extended Jacobson's original argument to the case where the horizon is in a non-equilibrium state. The basic result being that whereas the assumption of equilibrium leads to the Einstein equations (or equivalently the Einstein-Hilbert action), the assumption of deviations from equilibrium yields the Einstein-Hilbert action plus higher-order terms such as $R^2$, which would also arise as quantum corrections from any complete quantum gravity theory.
In addition there are the papers by Padmanabhan and Verlinde which set the physics world aflutter with cries of "entropic gravity". Then there is the holographic principle/covariant entropy bound/ads-cft which also suggest a thermodynamic interpretation of GR. As a simple illustration a black-hole in $AdS_5$ with horizon temperature $T$ encodes a boundary CFT state which describes a quark-gluon plasma at equilibrium at temperature ... $T$!
To top it all there is the very recent work Bredberg, Keeler, Lysov and Strominger - From Navier-Stokes To Einstein which shows an (apparently) exact correspondence between the solutions of the incompressible Navier-Stokes equation in $p+1$ dimensions with solutions of the vacuum Einstein equations in $p+2$ dimensions. According to the abstract:

The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70's and resurfaced recently in studies of the AdS/CFT correspondence.

To sum it all up let me quote from Jacobson's seminal 1995 paper:

Since the sound field is only a statistically defined observable on the fundamental phase space of the multiparticle sys- tem, it should not be canonically quantized as if it were a fundamental field, even though there is no question that the individual molecules are quantum mechanical. By analogy, the viewpoint developed here suggests that it may not be correct to canonically quantize the Einstein equations, even if they describe a phenomenon that is ultimately quantum mechanical. (emph. mine)


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A: I am very much surprised to see that apart from all the valid reasons (specially the argument, since everything else is quantum hence gravity should also be the same otherwise many inconsistencies will develop) mentioned by Lubos et. al. no body pointed out that one of the other main motivations to quantize gravity was that classical GR predicted singularities in extreme situations like big bang or black holes. It was kind of like the the instability of the Ratherford atomic model where electrons should have been spiral inward the nuclus as per the classical electrodynamics. Quantum theory saved physics from this obvious failure of classical physics. Naturally it occured to physicists that quantum theory should be the answer of the singularity problem of classical GR too. However experiences in the last 40 years have been different. Far from removing singularities it appears that our best quantum gravity theory is saying that some of the singularities are damn real. So obviously the motivation of quantization of gravity has changed to an extent and it is unification which is now driving the QG program in my humble opinion.
Some additional comments:
@Mbn, There are strong reasons to believe that the uncertainty principle is more fundamental than most other principles. It is such an inescapable property of the universe that all sane physicists imho will try their best, to make every part of their world view including gravity, consistent with the uncertainty principle. All of the fundamental physics has already been successfully combined with it except gravity. That's why we need to quantize gravity.
A: There are two questions here. The first is not so much whether we expect a unifying theory to be "quantum" as much as whether we expect a unifying theory to be probabilistic/statistical. I suppose that at or within 5 or 10 orders of magnitude of the Planck scale we can expect that we will still have to work with a statistical theory. Insofar as Hilbert space methods are the simplest effective mathematics for generating probability measures that can then be compared with the statistics of measurements, it's likely we will keep using this mathematics until some sort of no-go theorem proves that we have to use more sophisticated and harder to use mathematical tools (non-associative algebras of observables, etc., etc., etc., none of which most of us will choose to use unless we really have to).
The arguably more characteristic feature of quantum theory is a scale of action, Planck's constant, which determines, inter alia, the scale of quantum fluctuations and the minimal incompatibilities of idealized measurements. From this we have the Planck length scale, given the other fundamental constants, the speed of light and the gravitational constant. From this point of view, to say that we wish to "quantize" gravity is to assume that the Planck scale is not superseded in dynamical significance at very small scales by some other length scale.
The lack of detailed experimental data and an analysis that adequately indicates a natural form for an ansatz for which we would fit parameters to the experimental data is problematic for QG. There is also a larger problem, unification of the standard model with gravity, not just the quantization of gravity, which introduces other questions. In this wider context, we can construct any length scale we like by multiplying the Planck length by arbitrary powers of the fine structure constant, any of which might be natural given whatever we use to model the dynamics effectively. The natural length for electro-geometrodynamics might be $\ell_P\alpha^{-20.172}$ (or whatever, $\ell_P e^{\alpha^{-1}}$ isn't natural in current mathematics, but something as remarkable might be in the future), depending on the effective dynamics, and presumably we should also consider the length scales of QCD.
Notwithstanding all this, it is reasonable to extrapolate the current mathematics and effective dynamics to discover what signatures we should expect on that basis. We have reason to think that determining and studying in detail how experimental data is different from the expected signatures will ultimately suggest to someone an ansatz that fits the experimental data well with relatively few parameters. Presumably it will be conic sections instead of circles.
A: I will take a very simplistic view here.  This is a good question and was carefully phrased: «gravity ... be quantised ... ». Unification is not quite an answer to this particular question.  If GenRel produces singularities, as it does, then one can wonder if those singularities can really be the exact truth.  Since singularities have been smoothed over by QM in some other contexts, this is a motivation for doing that to GenREl which was done to classical mechanics and E&M.  But not necessarily for « quantising gravity ».  According to GenRel, gravity is not a force.  It is simply the effect of the curvature of space-time... In classical mechanics, the Coulomb force was a real force...  So if we are going to be motivated to do to GenRel that which was done to classical mechanics, it would not be natural to quantise gravity, but rather to formulate QM in a curved space-time (with the appropriate back-reaction---and that, of course, is the killer since probably some totally new and original idea is necessary here, so that the result will be essentially quantum enough to be a unification).  MBN has explicitly contrasted these two different options: quantising gravity versus doing QM or QFT in curved space-time.  Either approach addresses pretty much every issue raised here: either would provide unification.  Both would offer hopes of smoothing out the singularities.
So, to sum up the answer
IMHO there is no compelling reason to prefer quantising gravity over developing QFT in curved space-time, but neither is easy and the Physics community is not yet convinced by any of the proposals.
A: Actually, this reply is meant to address the specific (now closed) question:
Relativity and QM
Relativity and QM
and, to a lesser degree, the question (also now closed) that it was deferred to
Why are gravitons needed at all? [duplicate]
Why are gravitons needed at all?
It's a belief held by many in the Physics community, but not universally shared, that we need to "quantize" gravity.
The primary reason - if there is one - is that the alternative, of hybridizing a classical gravitational field (that is: the curved spacetime continuum itself) with a quantum system is open; because the general issue of hybridizing any kind of classical dynamics with quantum dynamics, whether and how it can even be done, is still an open issue.
The domain of "hybrid classico-quantum dynamics" is littered with a minefield of no-go theorems and impossibility results that rule out many, or most, of the natural intuitive ways one might try to accomplish this task. Diosi is a major player in this area (and many of his attempts have hit no-go mines), and it intersects with the area of "realistic wave function collapse" theories, since this too entails a kind of classico-quantum hybrid dynamics.
Wave Function Collapse
https://en.wikipedia.org/wiki/Wave_function_collapse
The problem alluded to here:
"More importantly, is not enough to explain actual wave function collapse, as decoherence does not reduce it to a single eigenstate."
is that the usual account of environment-induced "decoherence" gets us part of the way to explaining "wave function collapse", but not all the way. The underlying issue is known as "classicalization".
The two areas intersect on the hot-button issue of "gravitational decoherence"; and it may very well be that space-time induced decoherence could be that last missing key needed to resolve the issue of classicalization.
But not unless, or until a verifiable consensus is established on how to formulate hybrid classico-quantum dynamics.
The main obstacle to classico-quantum hybrid dynamics is that there is no consensus on whether or how a quantum sub-system can affect a classical sub-system in any hybrid dynamics. For a classico-quantum gravitational theory that means: in what way do the matter sources, described by quantum theory, bring about space-time curvature? That's the "back-reaction" problem.
On the left hand side of Einstein's equation $G_{μν} = -κ T_{μν}$ is a quantity $G_{μν}$ that is entirely geometric in nature, while on the right is a quantity $T_{μν}$ that is entirely matter-related and quantum in nature. There's a mismatch there.
Resolving that mismatch is the major open issue. Either the geometry is (somehow) "quantized" or the dynamics is (somehow) hybridized - or even a little bit of both. Take your pick.
If the geometry is quantized, meaning specifically that the modes of gravitational waves are quantized (hence, "gravitons"), then that means that the Weyl tensor, itself, which gives rise to those modes in spacetimes of dimension 4 or greater, must also be a quantized field.
The biggest problem with this is that the light cone field and the "causality" relation, itself, are given by the Weyl tensor; the question of whether you can even write down a quantum field theory or not appears to depend critically on the causal structure of space-time, so if the Weyl tensor were quantized, you could literally have the ability to form a superposition of two states where quantum theory exists in one, but not in the other. Or ... if that absurdity were somehow avoided, you could have the ability to form a superposition of states in which two events A and B are in a causal relation in one state, but not in the other. That's a back door to causality violation, that we might term "light cone tunneling".
The usual argument to evade this conclusion is that space-time is somehow "quantized" and that this (somehow) dodges the structural issues of space-time itself. That can never be anything more than an evasion.
Like all such arguments, the burden rests on those making it to show how: by literally devising a theory that reaches consensus that demonstrates that it can be done.
But, more to the point, it betrays a deep misunderstanding of what quantum theory, itself, is in relation to classical dynamics. It is not some kind of "total break" with classical dynamics, but presents itself as a continuous deformation of classical dynamics which, in the forward direction, is referred to as "quantization" and in the reverse direction as "the classical limit". This criterion is none other than what is called the Correspondence Principle, and is one of the axioms of any new paradigm (be it Relativity or Quantum Theory) that justifies stating that the theories established on the new paradigm continue to be affirmed by all the experiments and discoveries that were established on the older paradigm. Drop the Correspondence Principle, and you cut the moorings, and all that is old must be separately established anew in the new paradigm.
If the underlying geometry, in a quantized theory, is so radically different that all of what is said about the Weyl tensor, light cone fields and causality are rendered irrelevant, then this must continue to be the case even in the classical limit. Otherwise, you'll have a threshold issue: at what point in the "classical limit" does the break happen? If it happens at the end point, itself, then you have a tacit way to precisely measure the value of an otherwise-arbitrary continuous quantity, which is impossible in principle. If it happens short of the end-point, then you have a case of theories that are still on the "quantized" side of the "classical limit" arrow in which all the properties relevant to the classical limit continue to hold true; which demonstrates that the issues are not consequences of quantization, but actually independent.
In other words, a "quantized" geometry that somehow renders all the issues related to the Weyl tensor, light cones and causality irrelevant, must be the quantization of a classical theory other than what we now have; we're quantizing the wrong classical theory; and the discrepancy must be something that exists already on the classical level.
Another way we can see that the evasion is little more than an evasion is that even before we take any kind of classical limit, if the quantum theory indeed is a quantization of General Relativity (or, say, of Einstein-Cartan Theory), then the coherent states will still reflect the classical limit; and all the problems associated with the Weyl tensor will still be present in that context.
So, there is no escaping the issue.
Thus, in one direction you have the Classico-Quantum Hybridization issue, and in the other direction you have the Light-Cone-Tunneling issue.
Take your pick.
A: I will answer recasting the question as a thought experiment, based on the example proposed by Lubos;
1) a quantum object A in a superposition of two states separated by a distance $X$ somewhere in empty space
2) A has an associated gravity, with associated space-time curvature
3) now system B, will approach the region where A is found, and measure space-time curvature, but will not interact directly with A or its non-gravitational fields
4) now the system M (aka Measuring Apparatus) approaches the region where both A and B are found, and it will try to measure state correlation between A and B states
"gravity is quantum" potential outcome:
A and B are statistically correlated (entangled), supporting that B coupled with a linear superposition of gravitational fields
"gravity is classical" potential outcome:
A and B are uncorrelated quantum mechanically (a direct product of both densities), supporting that any substantial gravity field will collapse (this is basically what Penrose proposes as a mechanism for measurement collapse)
