Bell inequalities quantitatively separate classical from quantum correlations, serving as a cornerstone for experimental tests of entanglement and for device-independent approaches. The compilation of Bell tests enables such scenarios to be implemented without the requirement of physical spatial separation: instead of isolated laboratories, cryptographic protocols are used to constrain and coordinate the behavior of the participants. A key question is to understand the asymptotic quantum values of these compiled tests that is, the limiting correlations achievable when the players have unbounded local dimension and perform sequential measurements. Technically, I will present an operator-algebraic framework connecting sequential measurements with commuting-operator models. This includes the construction of universal C*-algebras generated by sequential PVM/POVM families, the characterization of asymptotic correlations through their representations, and the development of key technical tools (such as a chain rule for Radon–Nikodym derivatives of completely positive maps) that facilitate comparison between different strategies. Finally, I will outline our main results and open problems arising from the connection between cryptographic compilation of Bell tests, the limits of quantum correlations, and the structure of C*-algebras of sequential measurements.