Time Symmetry and the Local Curvature of Quantum Information: A Geometric Formulation of the Arrow of Time

We analyse the short-time structure of quantum evolution through the survival probability S (t) =Tr₀ (t), and show that its Taylor expansion defines three local invariants (, , j) that characterize the instantaneous arrow of time, geometric curvature, and higher-order dissipative behaviour of open quantum systems. For GKSL dynamics and a pure initial state, we prove =12\, ṠL (0), linking the local arrow of time directly to the initial linear entropy-production rate. We further show that unitary dynamics enforce exact conditional time symmetry, while dissipative generators break this symmetry at linear order, fully captured by the invariant. We establish a spectral closure theorem showing that for Markovian dynamics, the local invariants satisfy a finite-order ODE determined by the eigenvalues of the GKSL generator, enabling full trajectory reconstruction from local data. We extend the framework beyond the Markovian regime: for non-Markovian dynamics, (t) can become negative — signalling information backflow — and the monotonicity of the integrated information time (t) =₀ᵗ (s) \, ds provides an operationally accessible witness of Markovianity, detecting P-indivisibility. The violation of finite-order spectral closure serves as an independent measurable signature of memory effects. The framework connects quantum speed limits, information geometry, irreversibility, and non-Markovianity witnesses within a single geometric formulation of temporal asymmetry.