The matching problem has no small symmetric SDP

Yannakakis [27, 26] showed that the matching problem does not have a small symmetric linear program. Rothvoß [23] recently proved that any, not necessarily symmetric, linear program also has exponential size. It is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size.