quantum curvature If a state can be a superposition of energy states, and mass equals energy (special relativity), and mass curves space-time (general relativity), then could we say that space-time around a quantum system that is in a superposition of states is also in a "superposition of curvatures"?
 A: You would ask most physicists (except sir Roger Penrose) and they will tell you that you need planck scale energies to measure quantum gravity
I would dare to suggest instead a gravitational generalization of the schr¨odinger cat and the cavendish experiment mashup:


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*take in vacuum space, some mass $M$ of the same order as the one used by cavendish do estimate $G$. Now have some quantum system of two states coupled to a system (the tricky experimental part of the setup) that will provide a thrust to $M$ or not depending of the measured eigenvalue of the quantum system

*if the thrust system really does not decohere significantly with the environment, you should have the mass (just like the cat) in a superposition of states of different position. So the spacetime curvature must be in a superposition as well

*now place test masses nearby. Does the phase in the different eigenstates of $M$ affect the spacetime? well it does affect the electromagnetic field, otherwise we wouldn't see interference terms of light, so it should affect gravity as well. 

*We should see interference terms in the gravitational field. Not that hard to detect, if you think that Cavendish did this measurement (less the quantum superposition part) in 1797!!!
