Studying the dynamics of complex quantum systems out-of-equilibrium is a central problem in modern physics. This field encompasses fundamental questions, like the nature of thermalization and how it can be avoided by some quantum system as well as practical questions about properties of materials. Recently, experimental techniques like ultra-cold atom experiments and other platforms for quantum simulation emerged that can access out-of-equilibrium phases such as Floquet time crystals.

However, the theoretical and computational study of out-of-equilibrium phenomena remains challenging. From a computational perspective, the fundamental challenge in quantum many-body physics lies in the exponential number of parameters necessary to describe the wavefunction. If their spatial entanglement is low, wavefunctions can be approximated with relatively few parameters using tensor network techniques. Since equilibrium wavefunctions have low spatial entanglement, this aspect makes computations viable. However, when simulating dynamics, spatial entanglement grows rapidly with the evolution time. In this thesis I present a new approach to many-body dynamics building on Feynman and Vernon's influence functional and combining insights from the field of open quantum systems with matrix product state techniques.

We consider dynamics of a subsystem, and view the rest of the many-body system as a quantum environment. The environment's properties are encoded in the influence matrix on the space of trajectories. Treating the influence matrix as a ``wave function'' in the temporal domain, we introduce the concept of Temporal Entanglement which can be interpreted as the ``quantum memory'' of the bath. In several broad and relevant classes of systems, such as some chaotic systems, localized and integral systems, temporal entanglement exhibits favorable scaling. This allows the influence matrix to be efficiently compressed as matrix product state, opening the door to a new family of computational methods based on low temporal rather than spatial entanglement. Dynamical properties related to many-body localization and chaotic behavior are reflected in the influence matrix, allowing for analytical studies of these phenomena. I further show that this approach can be successfully applied to quantum impurity problems, where an interacting subsystem is coupled to environments consisting of free fermions. These models are of high practical significance as an ingredient in current algorithms used to study the properties of correlated materials