Polynomial commitment schemes (PCS) enable a prover to commit to a polynomial and later reveal evaluations with succinct, verifiable proofs. These schemes are undergoing a crucial shift from classical to post-quantum constructions as essential parts of contemporary cryptographic systems, such as Verkle trees and zk-SNARKs. The main scheme families, from pairing-based KZG and transparent Bulletproofs to lattice-based and hash-based post-quantum alternatives, are systematically compared in this structured review to analyze this evolution. Our analysis reveals a fundamental efficiency-security tradeoff: post-quantum alternatives offer quantum resistance at the expense of larger proofs and higher computational overhead, while classical schemes, which rely on quantum-vulnerable assumptions, offer optimal performance with constant-sized proofs. We identify recurring issues with proof conciseness, verification efficiency, and adaptive security by synthesizing research works. We also suggest specific research directions to close the performance gap between classical and quantum-resistant constructions. This paper provides a strategic roadmap and technical reference for creating useful post-quantum polynomial commitments. This review covers foundational schemes and their core trade-offs.