On the sum-of-squares degree of symmetric quadratic functions

We study how well functions over the boolean hypercube of the form f_k(x) = (|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_∞-norm as well as in l₁-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [19] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on l₁-approximation of f_k; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [12]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.