In order to solve the quadratic assignment problem of moderate and large size to optimality within a branch-and-bound framework, one needs to implement a ``good" lower bound. For us, a ``good" bound is the one that provides a promising compromise between its quality and computational time. Clearly, compromises should take into the consideration sizes of the problem instances. In this talk, we first introduce a new SDP-based eigenspace relaxation that turns to be good for problems of moderate size. Then, we compare our relaxation with SDP relaxations introduced by Zhao, Karisch, Rendl and Wolkowicz, and by Peng, Zhu, Luo and Toh. Our comparison results with the size-dependent choice of an appropriate relaxation that can be successfully used within a branch-and-bound framework.