Variational estimates provide quantitative information about sequences without revealing the parts that dominate their behaviour. They are therefore used in critical problems in harmonic analysis. In this project, we focus on three central critical settings from harmonic analysis: Firstly, we address the norm variation of multiple ergodic averages involving four or more commuting transformations, utilizing singular Brascamp-Lieb inequalities. Secondly, we explore nonlinear phase unwinding by examining pointwise variational convergence in Malmquist-Takenaka series. Finally, we seek to advance the application of nonlinear Fourier series theory to quantum signal processing.