In this talk, I will discuss recent progress in our understanding of the long-term behavior of large solutions to nonlinear Schr\”odinger-type equations. Dispersive estimates are a crucial tool in studying nonlinear Schr\”odinger equations, and examples include Strichartz estimates and local decay estimates. I will also cover a new technique for proving local decay estimates that allows for the establishment of Strichartz estimates (global in time) when the potential is quasi-periodic in time and localized in space, in five spatial dimensions. This talk is based on a series of collaborative works with Avy Soffer.