Parity is not in QAC0 (Moore's parity conjecture)

Claims: for every fixed depth $d$, every $c\ge1$ and every $0<\varepsilon\le1/2$, for all large $n$ no depth-$d$ circuit of one-qubit unitaries and unbounded-arity Toffoli gates on at most $n^c$ qubits, ancillas starting in $|0\rangle$ and one output qubit measured, computes parity with probability at least $1/2+\varepsilon$ on every input. Taking $\varepsilon=1/6$ gives bounded error $2/3$. The companion paper gives a second proof via regular trajectories and pruning; via Xu-Li reductions the same obstruction holds for strict majority. It does not cover circuits with intermediate measurements, postselection or primitive fanout gates, and gives no explicit size-depth tradeoff beyond polynomial size.

Claims: for every fixed depth $d$, every $c\ge1$ and every $0<\varepsilon\le1/2$, for all large $n$ no depth-$d$ circuit of one-qubit unitaries and unbounded-arity Toffoli gates on at most $n^c$ qubits, ancillas starting in $|0\rangle$ and one output qubit measured, computes parity with probability at least $1/2+\varepsilon$ on every input. Taking $\varepsilon=1/6$ gives bounded error $2/3$. The companion paper gives a second proof via regular trajectories and pruning; via Xu-Li reductions the same obstruction holds for strict majority. It does not cover circuits with intermediate measurements, postselection or primitive fanout gates, and gives no explicit size-depth tradeoff beyond polynomial size.