Abstract: In the near future, there will likely be special-purpose quantum computers with 50-70 high-quality qubits and controllable nearest neighbor couplings. In this talk, I’ll discuss general theoretical foundations for how to use such devices to demonstrate ​“quantum supremacy”: that is, a clear quantum speedup for *some* task, motivated by the goal of overturning the Extended Church-Turing Thesis (which says that all physical systems can be efficiently simulated by classical computers) as confidently as possible. I’ll then discuss some new work on how these sorts of experiments could be used to generate certified random bits, for use in cryptographic protocols and other applications.