This thesis studies the complexity, thermalization, and entanglement properties of quantum mechanical systems and the implications of their behavior in high energy and condensed matter models. In Chapter 2, we use the geometry of Lie groups to study the circuit complexity of quantum mechanical systems and develop the technology to analytically predict the appearance of the obstruction to the complexity growth of integrable systems. We simulate the spin and fermionic systems, in which the amount of chaos can be controlled, and conjecture the close relation between quantum complexity and thermalization. In Chapter 3, we apply quantum information theory to an evaporating black hole in Jackiw-Teitelboim gravity in the context of AdS/CFT as a quantum error correction protocol. By calculating the entropy measures using gravitational path integrals, we establish the criterion for an error in the Hawking radiation to be correctable and identify the transition in the reversibility of the error channel as the Page transition. In Chapter 4, we study a 1+1 dimensional random circuit with measurement and dissipation. We map this type of quantum system to a classical Ising model analytically and show that the entanglement dynamics can be understood as the thermodynamics of the domain wall formed in the spin model, as one of the spatial dimensions stretches. We then confirm the calculated time scales in the classical theory qualitatively with simulations of realistic qubit circuits. Overall, we present a holistic picture of quantum systems, from those evolving with pure unitary, to those evolving also with influence from the environment. The complexity of the theory and entanglement of the states exhibit universal behavior at early times (∼ O(N)) and distinct long-time behavior, determined by their thermalization properties in spin chains, toy models of black holes in AdS/CFT, and monitored random circuits with dissipation.