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.

First, we give a protocol by which a verifier can be convinced that a 3SAT

formula of size n is satisfiable, with constant soundness, given ~O(sqrt(n))

unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on

the existence of very short PCPs. Second, 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. Third, we prove the nonexistence of "perfect disentanglers" for

simulating multiple Merlins with one.