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 is forbidden while all others occur. No classical hidden Markov model reproduces it with fewer than 1+ sqrt n states, an unbounded separation whose only resource is one relative phase of π. With K amplitudes the same interference yields K(K + 1)/2 decay channels, and for positive weights and generic rates the classical cost is exactly K(K + 1)/2 states. Both are realized by a depth-three circuit whose parameter count does not grow with the register.