When has a quadratic statistics a constant regression on the sample mean?

Let $x_1,\dots,x_n$ be independent identically distributed non-degenerate random variables with finite second moment. Let $Q$ be a quadratic statistics with real coefficients and $\Lambda=x_1+\dots+x_n$. The condition $\mathbf E(Q\mid\Lambda)=\mathbf EQ$ is studied.