So there is no a full quantum theory of gravitation. However, there are instances where quantum effects due to gravitation have been studied. Like Gravitational neutron interferometry https://arxiv.org/abs/1701.00259 or maybe gravitational decoherence https://www.nature.com/articles/nphys3366
I would like to know if there are others quantum effects that involve gravity that can be studied (at least in an approximate way) without a full theory of quantized gravity.
If we could prepare a quantum state in which an object having significant mass is in a superposition of two different locations, and if the gravitational effects of that mass could be measured, then we'd be starting to explore quantum gravity experimentally even though such a situation can be described using a Newtonian $1/r^2$ model for the gravitational force. This can be studied without using stuff like QFT or string theory.
Some proposals for such experiments have been published recently because of some technical advances that are making them more nearly feasible. Here are a few examples of proposals and reviews of proposals:
https://arxiv.org/abs/1906.04513, "Testing the gravitational field generated by a quantum superposition"
https://arxiv.org/abs/1810.10202, "Searching for Signatures of Quantum Gravity in Quantum Gases"
https://arxiv.org/abs/1809.05081, "Demonstration of displacement sensing of a mg-scale pendulum for mm- and mg- scale gravity measurements"
https://arxiv.org/abs/1808.05842, "On the possibility of laboratory evidence for quantum superposition of geometries"
https://arxiv.org/abs/1807.11494, "Tabletop experiments for quantum gravity"
https://arxiv.org/abs/1807.07015, "Quantum Superposition of Massive Objects and the Quantization of Gravity"
https://arxiv.org/abs/1707.06050, "A Spin Entanglement Witness for Quantum Gravity"
In contrast to experiments like gravitational neutron interferometry, which can be described using a fixed spacetime background, the papers listed above are proposing ways to experimentally probe quantum superpositions of different gravitational fields.
If we stick with a Newtonian $1/r^2$ model of gravity, then describing a non-relativistic quantum system with gravitational interactions between the quantum particles is no problem at all: we can do this in the same way we handle the Coulomb interaction, like we often do in ab initio calculations in quantum chemistry. The obstacle to formulating relativistic quantum gravity is that our current understanding of relativistic quantum theory (namely quantum field theory) relies on having a prescribed spacetime background that, among other things, defines what "timelike separation" and "spacelike separation" mean. Usually when people talk about quantum gravity, they're talking about / hoping for a quantum theory that accounts for the dynamics of spacetime, as general relativity does classically.
Still, since we already know from macroscopic experiments that Newton's model of gravity is just an approximation to general relativity, an experiment like the proposals listed above would still provide important confirmation (or refutation?!?) of the expectation that the quantum superposition princple applies to the gravitational field, just like we already know it does to the electromagnetic field.
