Temporal Complexity Hierarchies in Solvable Quantum Many-Body Dynamics

The influence matrix (IM) provides a powerful framework for characterizing nonequilibrium quantum many-body dynamics by encoding temporal correlations into tensor-network states. In this talk, I will report recent progress toward a systematic understanding of the IM complexity for a class of quantum circuits ranging from integrable to chaotic regimes. By means of geometric group theory, I will demonstrate three distinct scaling behaviors of the IM temporal entanglement, establishing a hierarchy of computational resources required for accurate tensor-network representations. I will further analyze the memory structure of the IM and distinguish between classical and quantum correlations, identifying cases of classical IMs admitting Monte Carlo simulations, and introducing an operational measure of quantum memory for generic settings. These results establish a new connection between quantum many-body dynamics and group theory, providing fresh insights into the complexity of the IM and its connection to physical characteristics of the dynamics.