Incompatibility is a feature of quantum theory that sets it apart from classical theory, and the inability to clone an unknown quantum state is one of the most fundamental instances. The no-hiding theorem is another such instance that arises in the context of the black-hole information paradox, and can be viewed as being dual to no-cloning. Here, we formulate both of these fundamental features of quantum theory in a single form that is amenable to efficient verification, and that is robust to errors arising in state preparation and measurements. We extend the notion of unitarity—an average figure of merit that for quantum theory captures the coherence of a quantum channel—to general physical theories. Then, we introduce the notion of compatible unitarity pair (CUP) sets, that correspond to the allowed values of unitarities for compatible channels in the theory. We show that a CUP set constitutes a simple “fingerprint” of a physical theory, and that incompatibility can be studied through them. We derive information-disturbance constraints on quantum CUP sets that encode both the no-cloning/broadcasting and no-hiding theorems of quantum theory. We then develop randomized benchmarking protocols that efficiently estimate quantum CUP sets and provide simulations using IBMQ of the simplest instance. Finally, we discuss ways in which CUP sets and quantum no-go theorems could provide additional information to benchmark quantum devices.