Can quantum hardness be demonstrated? I am disturbed by assertions that this or that classical cryptographic primitive is "quantum resistant", but my understanding is that no one knows yet how to demonstrate quantum hardness, or even if such demonstrations are possible. At best, there may be cryptographic primitives for which no quantum attack has yet been found (but of course the world does not have much experience yet with quantum computers!)

Is this correct? Or, is there a rigorous way to demonstrate quantum hardness that I have not yet seen?
"Classical hardness" is demonstrated in the very same way, namely, we find a problem that is believed to be hard then we reduce the task of breaking the security of the scheme to the task of solving that problem.

So, it is kind of the same situation, because maybe there exist classical polynomial-time solutions to the problem used and those solutions simply were not discovered yet.

The main difference is that, in general, we are far more used to classical computers than to quantum ones and maybe saying that a problem is believed to be classical hard is more meaningful than the quantum analogous because we have really studied those problems a lot in the classical setting...