  A paramount topic in quantum foundations, rooted in the study of the EPR
paradox and Bell inequalities, is that of characterizing quantum theory in
terms of the space-like correlations it allows. Here we show that to focus only
on space-like correlations is not enough: we explicitly construct a toy model
theory that, while not contradicting classical and quantum theories at the
level of space-like correlations, still displays an anomalous behavior in its
time-like correlations. We call this anomaly, quantified in terms of a specific
communication game, the "hypersignaling" phenomena. We hence conclude that the
"principle of quantumness," if it exists, cannot be found in space-like
correlations alone: nontrivial constraints need to be imposed also on time-like
correlations, in order to exclude hypersignaling theories.
