  Information causality was initially proposed as a physical principle aimed at
deriving the predictions of quantum mechanics on the type of correlations
observed in the Bell experiment. In the same work, information causality was
famously shown to imply the Uffink inequality that approximates the set of
quantum correlations and rederives Tsirelson's bound of the
Clauser-Horne-Shimony-Holt inequality. This result found limited
generalizations due to the difficulty of deducing implications of the
information causality principle on the set of nonlocal correlations. In this
paper, we present a simple technique for obtaining polynomial inequalities from
information causality, bounding the set of physical correlations in any Bell
scenario. To demonstrate our method, we derive a family of inequalities which
non-trivially constrains the set of nonlocal correlations in Bell scenarios
with binary outcomes and equal number of measurement settings. Finally, we
propose an improved statement of the information causality principle, obtain
tighter constraints for the simplest Bell scenario that goes beyond the Uffink
inequality, and recovers a part of the boundary of the quantum set.
