Syndrome extraction for a CSS code has depth at least the Tanner-graph degree $Δ$, attained by edge colouring if gate order is ignored; correctness adds a parity constraint. We prove three general obstructions to depth-$Δ$ extraction and show that, for every two-block group-algebra code with weight-three polynomials whose checks overlap in at most two qubits, no correct single-round circuit with one ancilla per check and one CNOT per Tanner edge has depth six: the minimum is seven, so the seven-layer cycle of Bravyi \emph{et al.} is optimal for their seven bivariate bicycle codes. This is the first analytic, family-wide impossibility result, extending prior instance-specific solver-based infeasibility results. The hypothesis cannot be dropped: a single orbit of four-qubit overlaps with a collision element of even order and $2$-height at most eight makes depth six attainable, and a $[[72,8,6]]$ code of this kind has a verified six-layer circuit. The refutations carry DRAT certificates. As a further application of the same scheduling scheme, we construct Stab-QRAM, a quantum random access memory for affine Boolean data that uses only CNOT and $X$ gates and has the minimum depth possible for this gate set.
We study cooperative multiplayer bandits with adversarial rewards and no collision information. In each of $T$ rounds, each of $m$ labeled players picks one of $K>m$ arms. A player alone on its arm receives that arm's reward in $[0,1]$; players who share an arm receive $0$. Each player observes only its own reward, so it cannot distinguish a collision from an arm with zero reward, and there is neither shared randomness nor a communication channel. Bubeck, Li, Peres and Sellke (2020) obtained regret $\widetilde O(KT^{3/4})$ for two players and $\widetilde O(mK^{3/2}T^{1-1/(2m)})$ for $m$ players, and conjectured that regret $Ω(T^{2/3})$ is unavoidable already for two players. We refute this conjecture. We give a strategy whose expected regret against every oblivious adversary is $O(m^{3/2}\sqrt{KT}\log^2(KmT))$, which is optimal in $K$ and $T$ up to logarithmic factors. Three ideas make this possible. (i) Bits are sent through deliberate collisions as random parity checks, and the regret spent while a message is delayed is charged to the number of arms with positive reward rather than to elapsed time: an adversary that jams the channel with zero rewards forfeits the regret it could inflict. (ii) An exponential-weights learner over sets of arms combines a frequently resampled coordinator arm with a follower assignment that changes only $O(\sqrt{mT/K})$ times in expectation; delayed updates are reconstructed exactly. (iii) We teach the followers a privately chosen row index in a fixed public table using reward observations alone, allowing them to coordinate without shared randomness. The bound holds simultaneously for all reward matrices, even ones chosen with knowledge of the table, and a random table works with probability $1-O(1/(KmT))$.