This doctoral thesis analytically and numerically examines some of the most important concepts in quantum correlations in low-dimensional physics: entanglement and out-of-equilibrium dynamics. As John Bell once said: Entanglement expresses the spooky nonlocality inherent to quantum mechanics, and its study not only concerns the foundations of any quantum theory, but also has important applications in quantum information and condensed matter theory, amongst others. The first chapters are devoted to the study of entanglement entropies, a popular measure of the quantumness of a physical system....