As far as I know, there are no known observations that require a quantum theory of gravity. Now, that's not to say we don't need a quantum theory of gravity. In fact, I get the sense that we do, at least in part, because GR couples energy density to spacetime curvature, so we expect spacetime to be influenced by quantum fields, which begs for a quantum description of gravity. However, this doesn't imply that gravity is quantized. Gravity could, as far as I know, be classical, which I mean in the information-theoretic sense of the word: gravitational degrees of freedom cannot be entangled.
I am aware of Eppley-Hannah's paper that argues that gravity cannot be classical. Roughly, they show that if it were, then one could violate the uncertainty principle using a gravitational wave measurement device (which classically could have arbitrarily low momentum and wavelength). However, there are many rebuttals out there, including Mattingly's gr-qc/0601127. I came up with an alternative thought experiment, and I am interested to know what is wrong with it.
Suppose gravity is classical. Consider a box with a massive particle inside, in a state that makes it equally probable for it to be on the left or right side of the box. Now, partition the two sides with an impenetrable barrier. Then the wavefunction is $$ \psi(x) = \frac{1}{\sqrt{2}}\big(\psi_L(x) + \psi_R(x)\big), $$ where $\psi_L(x)$ and $\psi_R(x)$ are the same, except they only have support on their respective side of the box. Now, with your friend, Alice, separate the two sides of the box as far as you can before considerable decoherence ruins the superposition. I see two possibilities, based on whether gravity contributes to decoherence or not.
If gravity does contribute to decoherence, then the wavefunction must "collapse" before the boxes were separated (since the collapse was induced either by the particle's self-gravity or the acceleration imparted on the box when the boxes were first separated). But this conflicts with experiment, because we can create nonlocal superpositions that were initially local (e.g., in tests of entanglement).
If gravity does not contribute to decoherence, then, because gravity is classical, spacetime cannot warp under both boxes (and exist in a superposition like the particle that induces the warping). It can only warp under one box, for otherwise if Alice looks and does not find the electron, then spacetime would have to either suddenly unwarp to accommodate Alice's observation (which is nonclassical and would presumably send gravitational waves in all directions) or spacetime would remain warped under both boxes, despite there not being a source of energy to cause the warping around Alice's box. The only conclusion I see is that spacetime is warped under the box with the electron "in" it, which means that gravity "collapsed" the wavefunction early on --- a contradiction.
I'd like to know what is fundamentally flawed here. This argument seems too easy and obvious to rule out classical gravity. I suspect it's my assumption about how the energy density of the particle is allocated across spacetime pre-measurement. But I could see this as a non-issue, for is it not true that, however the energy density is allocated across spacetime pre-measurement, after Alice looks into her box to find it's (not) there, the spacetime must be warped under her (your) box, so my second argument above still applies?
