I will describe a few algorithmic advances which reduce computational bottlenecks in simulations of quantum many-body dynamics. In time-dependent density functional theory (TDDFT), the many-body wavefunction is approximated using a collection of single-particle wavefunctions, which independently satisfy the Schrödinger equation and are coupled through an effective potential. I will introduce a high-order, FFT-based solver for free space (nonperiodic) problems in TDDFT which sidesteps the usual requirement of imposing artificial boundary conditions. Many-body Green's functions, which describe correlations between quantum observables, enable practical simulations beyond the effective one-body picture of TDDFT. The Green's functions satisfy history dependent Volterra integro-differential equations with kernel nonlinearities. I will outline efficient history integration algorithms which significantly extend feasible propagation times in both equilibrium and nonequilibrium calculations. Speaker(s): Jason Kaye (Flatiron Institute)