QMA promise class with a verifier effect on witnesses

Let $\mathrm{\Pi }=(Y,N)$ be a promise problem, $0\le s(n)<c(n)\le 1$ efficiently encoded rational thresholds, and $w(n),a(n)$ polynomially bounded witness and zero-initialized workspace lengths. Require $c(n)-s(n)\ge 1\mathrm{/}p(n)$ for some polynomially bounded positive $p$. Then $\mathrm{\Pi }\in \mathrm{Q}\mathrm{M}\mathrm{A}(c,s)$ if there is a full-list uniform verifier interface as in Uniform verifier interface with a quantum witness such that every $x\in Y$ has a witness density operator $\rho$ with $\mathrm{Tr}(E_x\rho )\ge c(\mathrm{\mid }x\mathrm{\mid })$, and every $x\in N$ and every witness density operator satisfy $\mathrm{Tr}(E_x\rho )\le s(\mathrm{\mid }x\mathrm{\mid })$. Behavior outside $Y\cup N$ is unrestricted. The standard bounded-error class uses $c=2\mathrm{/}3$ and $s=1\mathrm{/}3$. The effect and probability are well-defined by A finite verifier induces an effect on its witness.