Summary: \(Q\)-reflexive Banach spaces were introduced by R. M. Aron and S. Dineen as those Banach spaces \(E\) such that all scalar continuous polynomials on \(E\) are weakly continuous on bounded sets and the second dual \({\mathcal P}({}^kE)^{**}\) of \({\mathcal P}({}^kE)\) can be identified in a canonical way with the space \({\mathcal P}({}^kE^{**})\). Here we present a different approach: in the definition of \(Q\)-reflexive spaces we do not ask the scalar polynomials in \(E\) to be weakly continuous on bounded sets. We give some conditions implying a space to be \(Q\)-reflexive. As a consequence, we can show an example of a \(Q\)-reflexive space \(F\) which is not quasi-reflexive; i.e., \(F^{**}/F\) is infinite-dimensional. Moreover, for Banach spaces \(E\) such that \(E^{**}\) has the approximation property, we show that our definition coincides with that of Aron and Dineen, and \(E^*\) contains no copies of \(\ell_p\) \((1\leq p< \infty)\) when \(E\) is \(Q\)-reflexive.