Ilya Vilkoviskiy: Temporal entanglement and the complexity of influence matrix in quantum chaotic circuits

The influence matrix (IM) provides a powerful framework for describing non-equilibrium quantum many-body dynamics by encoding multitime correlations into a tensor-network object. In this talk, I will review our recent results that bridge the scaling of temporal entanglement (TE), non-Markovianity, and computational complexity. Namely, I will clarify a seeming paradox between the observed volume-law TE, which suggests high simulation complexity, and the simplicity of correlation functions, which admits a compact MPS representation with a much smaller bond dimension than TE would naively imply. Additionally, I will introduce a family of solvable models that interpolate between integrable and chaotic regimes. These models display distinct behaviors of TE, complexity, and quantum memory. In particular, when quantum memory vanishes, the IM becomes classically simulable despite the underlying bulk dynamics remaining fully quantum.