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I'll also throw in a few words about the issue. As TwoBs points out, the problem is the comparative weakness of gravity, which makes quantum effects only relevant under extreme conditions: Essentially black holes and the big bang. The problem with the former is that we cannot peek beyond the horizon, so it might be more fruitful to try to tackle the issue by ...

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A quantum theory of gravity does make definite predictions. One such an example, which is the same for any theory of quantum gravity that reproduce GR at low energy, is the famous correction to the newton $1/r$ potential:  V(r)=\frac{M_{star}}{M_{Planck}r}\left(1-\frac{M_{star}}{M_{Planck}^2 r}-\frac{127}{30\pi^2}\frac{1}{M_{Planck}^2 r^2}+\ldots\right). ...

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I would say that there is not too much experimental evidence for a quantum theory of gravity yet, the reasons why such a theory is desirable are mainly of conceptual/theoretical nature. I will give a (likely to be incomplete) list of motivations for studying quantum gravity. Unification of all four fundamental interactions: The Standard Model of particle ...

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