Imagine I am doing Einstein's famous thought experiment: I'm in a lift and I'm asked to do local experiments to determine if I am accelerating in a non-inertial reference frame or in some gravitational potential accelerating me.
Now, I get clever and measure the standard deviation of acceleration. Standard deviation is a property of a quantum mechanical operator. This is a legal move see:
Can one define an acceleration operator in quantum mechanics?
And check if it mimics the standard deviation in a non-inertial reference frame:
The 3 fictitious forces of the rotating frame in Quantum Mechanics
By the equivalence principle they must be the same as otherwise I could tell you if I was locally in a gravitational field or not. Can I claim the quantization of the gravitational potential must also be the same? Note, one can imagine an array of of possible cases for example of the potential being $mg \hat x$ (where $m$ is mass, $g$ is the gravitational field and $x$ is the position operator)
