We advance the state-of-the art for zero-knowledge commit-and-prove SNARKs (CP-SNARKs). CP-SNARKs are an important class of SNARKs which, using commitments as “glue”, allow to efficiently combine proof systems—e.g., general-purpose SNARKs (an efficient way to prove statements about circuits) and Σ -protocols (an efficient way to prove statements about group operations). Thus, CP-SNARKs allow to efficiently provide zero-knowledge proofs for composite statements such as h= H(g ^x) for some hash-function H. Our main contribution is providing the first construction of CP-SNARKs where the proof size is succinct in the number of commitments. We achieve our result by providing a general technique to compile Algebraic Holographic Proofs (AHP) (an underlying abstraction used in many modern SNARKs) with special “decomposition” properties into an efficient CP-SNARK. We then show that some of the most efficient AHP constructions—Marlin, PLONK, and Sonic—satisfy our compilation requirements. Our resulting SNARKs achieve universal and updatable reference strings, which are highly desirable features as they greatly reduce the trust needed in the SNARK setup phase.