Single trusted qubit is necessary and sufficient for quantum realisation of extremal no-signaling correlations

Quantum statistics can be considered from the perspective of postquantum no-signaling theories in which either none or only a certain number of quantum systems are trusted. In these scenarios, the role of states is played by the so-called no-signaling boxes or no-signaling assemblages respectively. It has been shown so far that in the usual Bell non-locality scenario with a single measurement run, quantum statistics can never reproduce an extremal non-local point within the set of no-signaling boxes. We provide here a general no-go rule showing that the latter stays true even if arbitrary sequential measurements are allowed. On the other hand, we prove a positive result showing that already a single trusted qubit is enough for quantum theory to produce a self-testable extremal point within the corresponding set of no-signaling assemblages. This result opens up the possibility for security proofs of cryptographic protocols against general no-signaling adversaries.