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The interactive proof system model of computation has been studied extensively in computational complexity theory and theoretical cryptography for more than 25 years, and has driven the development of interesting new techniques and insights in those fields. This work considers the quantum interactive proof system model, which is the classical model's natural quantum computational analog. An exact characterization of the expressive power of quantum interactive proof systems is obtained: the collection of computational problems having quantum interactive proof systems consists precisely of those problems solvable with an ordinary classical computer using at most a polynomial amount of memory (or QIP = PSPACE in complexity-theoretic terminology). One striking implication of this characterization is that it implies quantum computing provides no increase in computational power whatsoever over classical computing in the context of interactive proof systems.