All Languages in NP Have Very Short Quantum Proofs

In this paper, we show that all languages in NP have logarithmic-size quantum proofs which can be verified provided that two unentangled copies are given. More formally, we introduce the complexity class QMA_log(2) and show that 3COL isin QMA_log(2). To obtain this strong and surprising result we have to relax the usual requirements: the completeness is one but the soundness is 1-1/poly. Since the natural classical equivalent of QMA_log(2) is uninteresting (it would be equal to P), this result, like many others, stresses the fact that quantum information is fundamentally different from classical information. It also contributes to our understanding of entanglement since QMA_log = BQP.