This position paper addresses the growing memory demands of multi-agent systems powered by large language models (LLMs). It frames multi-agent memory as a computer architecture problem, drawing parallels to traditional computer systems where memory hierarchy and bandwidth are critical bottlenecks. The authors distinguish between shared and distributed memory paradigms for agents and propose a three-layer memory hierarchy โ I/O, cache, and memory โ tailored for agentic systems. Key challenges identified include the need for protocols for cache sharing and memory access, and, crucially, establishing multi-agent memory consistency to ensure coherent and reliable operation.