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 O~(sqrt{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.