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The role played by parallelism in the theory of computation depends on the particular paradigm or computational environment considered, but its importance has been confirmed with the emergence of each novel computing technology. In this paper we study the implications of parallelism in quantum information theory and show that a parallel approach can make the difference between success and failure when trying to distinguish among (entangled) quantum states. A (perhaps surprising) consequence of this fact is the impossibility of constructing a Universal Computer, as defined herein. [ABSTRACT FROM AUTHOR]