Interactive proof systems are a classic idea in theoretical computer science, and have led to fundamental advances in complexity theory (hardness of approximation and the PCP theorem) and cryptography. Remarkably, in quantum information, interactive proof systems with multiple provers have become an important tool for studying quantum entanglement, extending the pioneering work of Bell in the 1960s. In this talk I will discuss recent progress in characterizing the power of the complexity class MIP* of such proof systems where the provers share entanglement. In addition to revealing an area of quantum complexity theory that is strikingly different from its classical counterpart, this work has led to new schemes for delegating quantum computations to untrusted servers, as well as to consequences for the Connes embedding problem in operator algebras. At the heart of this work are new protocols that use classical PCP techniques together with the rules of quantum mechanics to let a classical client precisely control an untrusted quantum server.
