Today’s most compact zero-knowledge arguments are based on the hardness of the discrete logarithm problem and related classical assumptions. If one is interested in quantum-safe solutions, then all of the known techniques stem from the PCP-based framework of Kilian (STOC 92) which can be instantiated based on the hardness of any collision-resistant hash function. Both approaches produce asymptotically logarithmic sized arguments but, by exploiting extra algebraic structure, the discrete logarithm arguments are a few orders of magnitude more compact in practice than the generic constructions. In this work, we present the first (poly)-logarithmic, potentially post-quantum zero-knowledge arguments that deviate from the PCP approach. At the core of succinct zero-knowledge proofs are succinct commitment schemes (in which the commitment and the opening proof are sub-linear in the message size), and we propose two such constructions based on the hardness of the (Ring)-Short Integer Solution (Ring-SIS) problem, each having certain trade-offs. For commitments to N secret values, the communication complexity of our first scheme is 𝑂̃ (𝑁1/𝑐) for any positive integer c, and 𝑂(log2𝑁) for the second. Both of these are a significant theoretical improvement over the previously best lattice construction by Bootle et al. (CRYPTO 2018) which gave 𝑂(𝑁‾‾√) -sized proofs.