A vector commitment scheme is a scheme (dough!) that allows a prover to prove that $v_i$ is a component of a vector $v$ without revealing any other information about $v$ . (So the prover commits to $v$ and then proves that $v_i$ is one of it's components).
To the best of my knowledge, current vector commitment schemes that use $O(1)$ space, rely on asymmetric cryptography, either polynomial commitments (depending on strong DH) or other schemes relying on CVP.
Is it known if vector commitment schemes can be done with symmetric crypto only? Are there any schemes or any impossibility results?