Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and a new, block-diagonal hierarchy is proposed. It has the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. It is applied to the stable set problem and experimental results for Paley graphs are reported.