Quantum pointer-parity round lower bound

There exist a real constant $C>0$ and a natural number $n_0\geq2$ such that, for every pair of natural numbers $k,n$ with $k>0$ and $n\geq n_0$, every starting vertex $v_1\in\{0,\ldots,n-1\}$, and every quantum protocol $P$ whose two inputs are maps $a,b:\{0,\ldots,n-1\}\to\{0,\ldots,n-1\}$ and whose outcome is Boolean, the following implication holds. Suppose $P$'s initial vector has norm $1$, it has exactly $k$ rounds with Bob speaking in even-numbered rounds and Alice in odd-numbered rounds, each speaker applies an input-dependent unitary acting within their currently owned wires and sends only wires they currently own, and ownership of sent wires passes to the other party. Suppose also that, for every input pair, the designated output party's input-dependent outcome matrices are positive semidefinite, sum to the identity, and act within that party's final owned wires. Acting within a wire set means being an arbitrary matrix on some injectively selected wires in that set tensored with the identity on all remaining wires; selecting zero wires is allowed. Define the successor on two disjoint copies of $\{0,\ldots,n-1\}$ by $L(v)\mapsto R(a(v))$ and $R(v)\mapsto L(b(v))$, apply it exactly $k+1$ times to $L(v_1)$, and label $L(v)$ by $v$ and $R(v)$ by $n+v$. If, for every $a,b$, the probability of outputting whether the sum of the binary digits of this label is odd is at least $2/3$, where the probability is the real trace of the corresponding outcome matrix times the final pure-state matrix after all round unitaries, then

The constant and threshold are common to all $k,n,v_1,P$; empty sent sets and arbitrary initial ownership assignments are permitted subject to the stated conditions.