Polynomial Commitment Schemes from Classical Constructions to Post-Quantum Directions

Polynomial commitment schemes (PCS) enable a prover to commit to a polynomial and later reveal evaluations with succinct, verifiable proofs. As critical components of modern cryptographic systems like Verkle trees and zk-SNARKs, these methods are experiencing a significant transition from classical to post-quantum designs. This comprehensive research systematically compares the major scheme families to examine this progression, from pairing-based KZG and transparent Bulletproofs to lattice-based and hash-based post-quantum alternatives. We present a unified taxonomy that maps the classical-to-post-quantum transition across trust models, security assumptions, and efficiency measures after conducting a PRISMA-guided systematic review of 77 works. Our analysis reveals a fundamental trade-off between efficiency and security: classical schemes, which rely on quantum-vulnerable assumptions, provide optimal performance with constant-sized proofs, while post-quantum alternatives offer quantum resistance at the cost of larger proofs and higher computational overhead. By combining research works, we highlight recurrent problems with adaptive security, verification efficiency, and proof conciseness. We offer a specific research roadmap with prioritized short-, medium-, and long-term directions to close the performance gap between quantum-resistant and classical architectures based on our quantitative analysis. This study offers a technical reference and a strategic roadmap for constructing practical post-quantum polynomial commitments.