At non-equilibrium, quantum systems exhibit rich phenomena. In this talk, I will show that some non-equilibrium dynamics, which appear distinct from each other, are governed by the same underlying hyperbolic geometry. For instance, the dynamical instability, expansions of unitary fermions released from harmonic traps, and some mysterious behaviors of breathers of Bose-Einstein condensates are different manifestations of the hyperbolic surface. The exceptional point that represents the PT symmetry breaking in non-Hermitian systems is also captured by the same geometry, being the light cone of an Anti-de Sitter spacetime. This geometric approach thus unifies a broad range of Hermitian and non-Hermitian quantum dynamics and provides experimentalists with a new tool to control quantum systems.