Several results on the monotone circuit complexity and the conjunctive complexity, i.e., the minimal number of AND gates in monotone circuits, of quadratic Boolean functions are proved. We focus on the comparison between single level circuits, which have only one level of AND gates, and arbitrary monotone circuits, and show that there is a huge gap between the conjunctive complexity of single level circuits and that of general monotone circuits for some explicit quadratic function. Almost tight upper bounds on the largest gap between the single level conjunctive complexity and the general conjunctive complexity over all quadratic functions are also proved. Moreover, we describe the way of lower bounding the single level circuit complexity, and give a set of quadratic functions whose monotone complexity is strictly smaller than its single level complexity.
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