Inspired by an idea due to Levenshtein, we apply the low correlation zone constraint in the analysis of the weighted mean square aperiodic correlation. Then we derive a lower bound on the measure for quasi-complementary sequence sets with low correlation zone (LCZ-QCSS). We discuss the conditions of tightness for the proposed bound. It turns out that the proposed bound is tighter than Liu-Guan-Ng-Chen bound for LCZ-QCSS. We also derive a lower bound for QCSS, which improves the Liu-Guan-Mow bound in general.