can solve our (artificially simplified) problem: in the $\mathsf{QMA/poly}$ protocol for $L$, we use certificates which isolate $f_{1},\ldots,f_{m}\in S$ as above as the classical advice for Arthur.  Arthur requests from Merlin each of the $m$ states $\xi_1, \ldots, \xi_m$ such that $f_{i}=f_{\xi_i}$, and verifies that he receives appropriate states by checking them against the certificates.  This involves multiple measurements of each $\xi_i
$—and an immediate difficulty is that, since measurements are irreversible in quantum mechanics, the process of verifying the witness state might also destroy it.   However, we get around this difficulty by appealing to a result of Aharonov and Regev [@ar].  This result essentially says that a $\mathsf{QMA}$ protocol in which Arthur is granted the (physically unrealistic) ability to perform ‘non-destructive measurements’ on his witness state, can be efficiently simulated by an ordinary $\mathsf{QMA}$ protocol.

To build intuition, we will begin (in Section \[MAJCER\]) by proving the majority-certificates lemma for Boolean functions, as described above.  However, to remove the artificial simplification we made and prove Theorem \[mainthm1\], we will need to generalize the lemma substantially, to a statement about possibly-infinite sets of real-valued functions $f:\left\{
0,1\right\}  ^{n}\rightarrow\left[  0,1\right]  $.  In the general version, the hypothesis that $S$ is finite 