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Mar 12, 2015 at 2:42 comment added Suvrit As I also mentioned in my comments, this problem in the exact incarnation of the OP also goes under the name MaxQP in the theory CS literature, and has been the subject of fairly intense study (it contains the MaxCUT problem as a special case), including bounds obtained from Grothendieck's inequality, etc.
Mar 12, 2015 at 2:26 comment added KPU Thank you Igor! I was wondering what the answer is when $n=3.$ Thanks!
Mar 12, 2015 at 1:36 history answered Igor Rivin CC BY-SA 3.0