*that* structure, as opposed to any other?

One way to approach this question, or at least gain some insight, is to compare and contrast quantum theory with other models - theoretical possibilities which do not describe our universe, but which can nonetheless be explored. In this paper, we investigate one particular non-classical, non-quantum theory. In [@barrett], this theory was called *generalized non-signalling theory*, or GNST for short, as it admits all non-signalling correlations [@popescu; @boxes]. Here we call it *box world*.

One of the notable features of box world is that, as in quantum theory, measurements on separate but entangled systems can produce outcomes that are nonlocally correlated, i.e., which cannot be explained by any local hidden-variable model [@bell; @CHSH]. It is already known that nonlocal correlations are useful in many information theoretic tasks [@qi1; @qi2]. But in quantum theory, there is a quantitative limit on the amount of nonlocality that correlations can have [@tsirelson]. In box world, by contrast, arbitrary nonlocal correlations can be produced, as long as they do not permit instantaneous signalling. This has consequences. For example, with stronger than quantum correlations, it is known that communication complexity problems become trivial, requiring only constant communication [@van-dam; @brassardetal]. 