  Correlations that violate a Bell Inequality are said to be nonlocal, i.e.
they do not admit a local and deterministic explanation. Great effort has been
devoted to study how the amount of nonlocality (as measured by a Bell
inequality violation) serves to quantify the amount of randomness present in
observed correlations. In this work we reverse this research program and ask
what do the randomness certification capabilities of a theory tell us about the
nonlocality of that theory. We find that, contrary to initial intuition,
maximally nonlocal theories cannot allow maximal randomness certification. We
go on and show that quantum theory, in contrast, permits certification of
maximal randomness in all dichotomic scenarios. We hence pose the question of
whether quantum theory is optimal for randomness, i.e. is it the most nonlocal
theory that allows maximal randomness certification? We answer this question in
the negative by identifying a larger-than-quantum set of correlations capable
of this feat. Not only are these results relevant to understanding quantum
mechanics' fundamental features, but also put fundamental restrictions on
device-independent protocols based on the no-signaling principle.
