  Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.
