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This paper focuses on the problem of learning compressed state representations for multi-agent tasks. Under the assumption of rich observation, we pinpoint that the state representations should be compressed both spatially and temporally to enable efficient prioritization of task-relevant features, while existing works typically fail. To overcome this limitation, we propose a novel method named Spatio-Temporal stAte compRession (STAR) that explicitly defines both spatial and temporal compression operations on the learned state representations to encode per-agent task-relevant features. Specifically, we first formalize this problem by introducing Task Informed Partially Observable Stochastic Game (TI-POSG). Then, we identify the spatial representation compression in it as encoding the latent states from the joint observations of all agents, and achieve this by learning representations that approximate the latent states based on the information theoretical principle. After that, we further extract the task-relevant features of each agent from these representations by aligning them based on their reward similarities, which is regarded as the temporal representation compression. Structurally, we implement these two compression by learning a set of agent-specific decoding functions and incorporate them into a critic shared by agents for scalable learning. We evaluate our method by developing decentralized policies on 12 maps of the StarCraft Multi-Agent Challenge benchmark, and the superior performance demonstrates its effectiveness.