The path integral formulation of quantum mechanics provides a powerful and vast set of tools for physicists and chemists to compute thermodynamic and dynamic quantities of condensed phase systems [1]. Unfortunately, path inte- gral calculations for large systems have high computational cost and in some cases converge poorly owing to the infamous sign problem and other numeri- cal instabilities. Developing new methods to more efficiently perform path in- tegral calculations in the condensed phase and exploring their numerical stabil- ity and convergence properties will be useful to many scientists who practice these methods. In this dissertation we discuss three methods to boost the per- formance of various path integral calculations, two of which rely on classical computing while the third utilizes quantum computing.First, we demonstrate how the ring polymer molecular dynamics (RPMD) framework for condensed phase systems can be made more efficient by using mixed time slicing (MTS) to reduce the Trotter dimension of the sub-system describing the environment [2]. In doing so, we derive a consistent method for introducing fictitious mass within the RPMD framework for a Hamiltonian containing multiple sub-systems. Next, we study mapping variable ring polymer molecular dynamics (MV- RPMD), a path integral method for non-adiabatic dynamics, and apply MTS to reduce the Trotter dimension of electronic or nuclear variables. With the ability to selectively reduce the dimensionality of the MV-RPMD Hamilto- nian, we study the ability to compute position auto-correlation functions. We also undertake the first study of the emergence of stable and unstable mani- folds of a Hamiltonian describing non-adiabatic dynamics using the method of Lagrangian descriptors[3, 4]. All MV-RPMD calculations are executed using MAVARIC, a software package developed within the Ananth group. A tutorial for using MAVARIC is provided toward the end of this dissertation. Finally, we develop a hybrid quantum-classical algorithm for running path integral Monte Carlo (PIMC) calculations in which complex time propagator matrix elements are computed in a position basis by calling a quantum com- puter. This method can be extended to system-bath models via the use of an in- fluence functional. We compute the thermal position auto-correlation function of a simple harmonic oscillator using quantum simulators as proof-of-principle and compare against exact results. Additionally, we show how our hybrid algo- rithm can be modified for studying the dynamics of spin systems.