It's common to see people oversimplify by saying that physics currently lacks the tools to describe any situation involving both quantum mechanics and gravity. Clearly this is not the case. For example, the Pound-Rebka experiment involves inescapably quantum-mechanical phenomena such as the Mossbauer effect, but is really just a test of the equivalence principle. Less trivially, Colella et al. did an experiment demonstrating interference between two beams of neutrons that had traveled through different gravitational potentials. It seems to me that there are probably a bunch of different levels of difficulty we could consider:
Experiments, such as the ones above, involving quantum mechanics, that can be described in flat spacetime using the equivalence principle. Curvature of spacetime is negligible.
Experiments in which curvature is nonnegligible, but the analysis is still trivial. For example, I could imagine, at least in principle, doing gravitational lensing with neutrons and observing quantum interference effects between different parts of the beam. (In reality, I'd guess this example wouldn't work due to decoherence.)
Semiclassical gravity, e.g., Hawking radiation.
Can anyone comment on whether this 4-level classification seems right, or give a more rigorous set of criteria for distinguishing the levels?