Standard quantum information and quantum foundations have historically focused on Bell non-locality in the simplest setting: bipartite, non-communicating systems. This thesis explores the boundaries of this standard Bell scenario by extending it along three main axes: restricted communication, multipartite settings, and operator-algebraic formulations.
To preserve the gap between classical, quantum and 'supra'-quantum correlations while allowing for information exchange, we investigate communication networks restricted by dimensional, computational (via homomorphic encryption), and operational constraints. In doing so, we develop rigorous analytical frameworks to handle the complex structure of multipartite correlations, focusing on sequential communication models and composable simultaneous purifications.
To tackle these challenges, the thesis introduces novel mathematical tools from the operator-algebraic framework, such as a new chain rule for the Radon-Nikodym Theorem. By bridging physics, computer science, and mathematics, this work provides both practical certification techniques for quantum devices and deep structural insights into the nature of physical correlations.