Update: A slightly different version of this question has been answered here.
As far as I can see, a major issue with Google's recent quantum supremacy claim is that it is hard to verify the results. If the quantum computer would be powerful enough to solve problems in NP (like factoring, using Shor's algorithm), it would be easy to verify the results, however, this computational power is not available yet. As an intermediate step,
what could be a problem within reach of today's quantum computers whose output can be verified efficiently with a witness?
Here efficiently need not stand for the problem itself being in NP, but rather it would be sufficient that the correctness of the solution is in NP. For example, consider a satisfiability task whose literals are $x_i$'s and $y_j$'s. If the quantum computer can produce all the $x_i$ that are part of a solution, then a witness for its correctness would be finding the $y_j$'s that make $(x,y)$ into a solution. The corresponding $y$ can be found with a classical algorithm, or set as a challenge, like in case of Bitcoin, to mint some quantum coin, with the computational power of all the miners looking for the witness. In fact, miners can be also rewarded for finding another $(x',y')$ solution of the original problem, which would determine whether $x$ is useful or not.
So what could be such a problem? Can Shor's algorithm or some other similar problem be broken into finding some $x$ with a quantum computer that can be extended into a witness with some (smaller) $y$?