The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs.   Many of the simplest questions about this class have remained embarrassingly open:   for example, can we give any evidence that k quantum proofs are more powerful than one?   Does QMA(k) = QMA(2)   for k ≥ 2?   Can QMA(k)   protocols be amplified to exponentially small error?

In this paper, we make progress on all of the above questions.

• We give a protocol by which a verifier can be convinced that a 3Sat formula of size m is satisfiable, with constant soundness, given Õ(√m) unentangled quantum witnesses with O(log m) qubits each.   Our protocol relies on the existence of very short PCPs.
• We show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2)   protocol can be amplified to exponentially small error, and QMA(k) = QMA(2)   for all k ≥ 2.
• We prove the nonexistence of   “perfect disentanglers”   for simulating multiple Merlins with one.