Workspace-Bounded Quantum Pointer Chasing

Pointer chasing measures how information propagates through multiple rounds of communication. Quantum versions of the problem have never incorporated bounded local memory, even though every realistic protocol operates under finite workspace. This paper defines a workspace-bounded quantum pointer-chasing model, where each party has at most S qubits of reusable memory and total communication T. A multi-round form of the Kadison--Schwarz packing lemma shows that bounded workspace limits distinguishable state evolution across k rounds, giving T\( \sqrt{S} \;\ge\; \Omega\!\big(k\sqrt{n}\big) \). The bound recovers T \( \ge \Omega(k\sqrt{n}) \) when S=1 and becomes trivial at T \( \ge \Omega(k) \)when S \( \ge \) n. It extends the single-round framework established in Workspace Bound and provides the first explicit multi-round tradeoff between communication and local memory in quantum protocols.