Memory Separations from a Collective-Spin Quantum Register

A spin one-half driven only by total-spin operators generates a renewal process in which one interevent time $n^{\ast }$ is forbidden while all others occur. No classical hidden Markov model reproduces it with fewer than $1+\sqrt{n^{\ast }}$ states, an unbounded separation whose only resource is one relative phase of $\pi$. With $K$ amplitudes the same interference yields $K(K+1)\mathrm{/}2$ decay channels, and for positive weights and generic rates the classical cost is exactly $K(K+1)\mathrm{/}2$ states. Both are realized by a depth-three circuit whose parameter count does not grow with the register.