  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.
