What would such an experiment look like? If it makes it easier, perhaps an experiment that's even harder (or much, much harder) to realize but easier to explain.
It's pretty simple. You basically have two suspended mirrors on wires and you shoot lasers at each mirror. Due to gravity there will be an interaction between the mirrors. There are then two things you can probe. First, the correlation between the two reflected lasers, and if you can achieve the extreme temperature requirements, entanglement between the two reflected lasers caused by the gravity. Now, the former doesn't really imply that gravity is quantum, it just shows that quantum correlations can be mediated by gravity, but this may also be true classically, however, the much more difficult to show entanglement would definitely require quantum gravity. The first step is still good though because no one has yet managed to demonstrate quantum correlations via gravity. The experiment would also have two mirrors two turn the laser beams into cavities, increasing the interaction strength. The setup would look a bit like this:
_ _
! !
---|-----| |-----|---
where | are mirrors, ! are suspension wires, and - is the laser beam
Those reflected beams would at best be very very weakly entangled, right? I'm not sure what's the name for it, but if you arranged the quantum state of the two beams into a 2x2 matrix then the determinant would be just a tiny bit non-zero.
