The class $\mathsf{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 $\mathsf{QMA}( k) =\mathsf{QMA}( 2) $ for $k\geq2$? Can $\mathsf{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 $\widetilde{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 $\mathsf{QMA}( 2) $ protocol can be amplified to exponentially small error, and $\mathsf{QMA}( k) =\mathsf{QMA}( 2) $ for all $k\geq2$.
• We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.