**Abstract:** It is well known that in the language of no-signaling boxes – consistent families of probabilities representing measurement correlations – the following three sets: local deterministic, quantum and no-signalling constitute the increasing sequence with strict inclusions. Moreover the outer set of no-signalling boxes has no nontrivial (ie. not local deterministic) extremal points that are generated by measurement on quantum states. This fact significantly complicates all quantum information security protocols potentially robust against attacks of more powerful, post-quantum adversaries.

In the case of no-signalling assamblages which attracted attention recently, the strict inclusion of the three analogous sets is also true except bipartite case. However the issue of extremality, potentially vital to information security, has been open for a long time.

We present results that fill gaps in the above picture. First, even in the sequential measurement scenario no nontrivial quantumly generated extremal point can exist in the set of no-signalling boxes. Second, somewhat surprisingly, the analog of this no-go theorem fails for no-signalling assamblages. This is for the first time when quantum mechanics is observed to produce points that are extremal in some post-quantum framework. The new concept of the *inflexibility of assamblages with pure quantum elements* plays here an important role.

We also study the boundary of the no-signalling assamblages. In analogy to quantum entanglement theory we define and study the edge of the set of assamblages. We found that here quantum mechanics can produce edge assamblages * via* measurements on at most rank-two 3-qubit states as opposed to rank five for the edge of the 3-qubit quantum entanglement set.

Future perspectives of the presented research will be discussed.