Lifting sum-of-squares lower bounds: degree-2 to degree-4

The degree-4 Sum-of-Squares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree-4 SoS SDP are not well understood. For example, existing integrality gaps do not rule out a (2−)-algorithm for Vertex Cover or a (0.878+)-algorithm for MaxCut via degree-4 SoS SDPs, each of which would refute the notorious Unique Games Conjecture.