Summary: The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form \(\mathbf{x}^T Q \mathbf{x}\) on the standard simplex \(\{\mathbf{x} \ge \mathbf{0} : \mathbf{x}^T \mathbf{e} = 1\}\), is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix \(Q = Q^T, Q_{i, j}, (i \le j)\) being independent and identically concave-distributed, attains its minimum at a point \(\mathbf{x}\) with support size \(|\{j : x_j > 0\}|\) bounded in probability. Later Chen and Peng proved that for \(Q = (M + M^T)/2\), with \(M_{i, j}\) i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in \(n\), the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint \(-\infty\), provided that the distribution of \(Q_{i, j}, (i \le j)\) (of \(M_{i, j}\), if \(Q = (M + M^T)/2\) resp.) has a (super/sub) exponentially narrow left tail.