In this paper, we investigate the relation between the QQ-spectrum and the structure of GG in terms of the circumference of GG. Exploiting this relation, we give a novel necessary condition for a graph to be Hamiltonian by means of its QQ-spectrum. We also determine the graphs with exactly one or two QQ-eigenvalues greater than or equal to 22 and obtain all minimal forbidden subgraphs and maximal graphs, as induced subgraphs, with respect to the latter property.