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Conditional lower bounds on the spectrahedral representation of explicit hyperbolicity cones

Over the past decade there has been growing interest on characterizing which convex cones over Rn are spectrahedral, that is, are a linear section of the cone of positive semidefinite matrices. This interest is largely motivated by applications in control theory, optimization and combinatorics. One particular class of convex cones of interest is the class of hyperbolicity cones, where the (still open) Generalized Lax Conjecture states that every hyperbolicity cone is spectrahedral. Recent works [1, 2] have established that the hyperbolicity cones of the elementary symmetric polynomials and the homogeneous multivariate matching polynomial are spectrahedral, but the question of whether there exists an efficient spectrahedral representation for such cones remains open. Previous work [11] has provided exponential lower bounds on the spectrahedral representation of non-explicit hyperbolicity cones which are known to be spectrahedral. The current best lower unconditional bounds for explicit cones are the linear lower bounds proved by [7]. In this paper we establish the first superpolynomial hardness of the minimal spectrahedral representation for an explicit family of hyperbolicity cones, assuming Valiant's VP vs VNP conjecture is true, that is, that the permanent polynomial cannot be computed by algebraic circuits of polynomial size. More precisely, we prove that the hyperbolicity cone of Amini's homogeneous matching polynomial must require superpolynomial spectrahedral representations, assuming that Valiant's conjecture is true. This is the first work providing a (conditional) superpolynomial lower bound on the spectrahedral representation of an explicit hyperbolicity cone.