© 2016 American Physical Society. Several information measures have recently been defined that capture the notion of recoverability. In particular, the fidelity of recovery quantifies how well one can recover a system A of a tripartite quantum state, defined on systems ABC, by acting on system C alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the type I error is constrained to be no larger than a constant.