An important consequence of special relativity, in particular, of the constant and finite speed of light, is that space-like separated regions in spacetime cannot communicate. This fact is often referred to as the “no-signaling principle” in physics.
However, even when signaling is in fact possible, there still are obvious constraints on how signaling can occur: for example, by sending one physical bit, no more than one bit of information can be communicated; by sending two physical bits, no more than two bits of information can be communicated; and so on. Such extra constraints, that by analogy we call “no-hypersignaling,” are not dictated by special relativity, but by the physical theory describing the system being transmitted. If the physical bit is described by classical theory, then the no-hypersignaling principle is true by definition. It is not so in quantum theory, where the validity of the no-hypersignaling principle becomes a non-trivial mathematical theorem relying on a recent result by Péter E. Frenkel and Mihály Weiner (whose proof, using the “supply-demand theorem” for bipartite graphs, is very interesting in itself).
As one may suspect, the no-hypersignaling principle does not hold in general: it is possible to construct artificial worlds in which the no-hypersignaling principle is violated. Such worlds are close relatives of the “box world,” a toy-model theory used to describe conceptual devices called Popescu-Rohrlich boxes. Exploring such alternative box worlds, one further discovers that the no-hypersignaling principle is logically independent of both the conventional no-signaling principle and the information causality principle, however related these two may seem to be with no-hypersignaling.
This means that the no-hypersignaling principle needs to be either assumed from the start, or derived from presently unknown physical principles analogous to the finite and constant speed of light behind Einstein’s no-signaling principle.