We present optimized and easy to follow numerical simulations for complex many-body quantum systems implemented in MATLAB. The examples include the calculation of the magnetization dynamics for the closed and open Ising model, dynamical quantum phase transition in cavity QED arrays, Markovian dynamics for interacting two-level systems, and the non-Markovian dynamics of the pure-dephasing spin-boson model. This tutorial will be a useful toolbox for undergraduate and graduate students with an interest in numerical simulations with applications in quantum optics and condensed matter problems.