We present improved versions of the standard semi-definite relaxation for quadratic programming, that underlies many important results in robustness analysis and combinatorial optimization. It is shown that the proposed polynomial time convex conditions are at least as strong as the standard ones, and usually better, but at a higher computational cost. Several applications of the new relaxations are provided, including less conservative upper bounds for the structured singular value CL and enhanced solutions for the MAX CUT graph partitioning problem.