One parity protocol family distills quantum and post-quantum boxes

There exists a family $W=(W_n)_{n\in\mathbb N}$ of protocols, where an $n$-copy protocol consists of Boolean input functions $\alpha,\beta:\{0,1\}\times\{0,\ldots,n-1\}\to\{0,1\}$ and Boolean output functions $f,g:\{0,1\}\times\{0,1\}^n\to\{0,1\}$, such that for every natural number $n$, including $0$ and $1$, $W_n$ is exactly the protocol $\alpha(x,i)=x$, $\beta(y,i)=y$, $f(x,u)=\bigoplus_i u_i$, and $g(y,v)=\bigoplus_i v_i$, where $\bigoplus$ is addition modulo two and the empty parity is $0$. There exist a natural number $n>1$ and a real Boolean-indexed array $Q$ for which $Q$ is quantum and $W_n$ strictly distills $Q$; there also exist a natural number $m>1$ and a real Boolean-indexed array $R$ for which $R$ is not quantum and $W_m$ strictly distills $R$. These two copy counts are separate existential choices. Here, for any such array $S$, its output under $W_k$ is

Writing $C_S(x,y)=S(x,y,0,0)+S(x,y,1,1)-S(x,y,0,1)-S(x,y,1,0)$ and $L(S)=\max_{x,y}|C_S(x,y)+C_S(x,1-y)+C_S(1-x,y)-C_S(1-x,1-y)|$, strict distillation means that both $S$ and $T_kS$ have nonnegative entries, total mass one at each input pair, first-output marginals independent of the second input and second-output marginals independent of the first input, and satisfy $2<L(S)<L(T_kS)$. An array $S$ is quantum precisely when there exist complete complex inner-product spaces $A,B$, a unit vector $\psi\in\widehat{A\otimes_{\mathbb C}B}$, and positive continuous complex-linear operators $M_{x,a}$ and $N_{y,b}$, summing to the identity over $a$ and $b$, respectively, for each input, such that $S(x,y,a,b)=\operatorname{Re}\langle\psi,\widehat{M_{x,a}\otimes N_{y,b}}\psi\rangle$ for every Boolean quadruple, with the hat denoting the induced map on the completion. The claim about $R$ denies the existence of any such representation.