The probabilistic nature of quantum theory has deep consequences both on our understanding of the theory and on the potential applications that can be built with it. In fact, quantum statistics play a key role throughout quantum information theory. Yet, quantum correlations remain poorly understood. While some methods have been proposed to approach these statistics, they are generally either partial or implicit. The aim of this internship is to identify some explicit regions of the quantum boundary analytically. By providing new ways of testing whether some statistics admit a quantum explanation or not, this work will help to characterize both the power and the limitations of quantum predictions. It may also lead to applications in self-testing and the certification of quantum technology devices, as well as in the study of quantum networks.