Blockchain technology has produced effective solutions and provides security by using cryptographic tools for various applications, attracting attention from the academic community. Therefore, researchers have taken advantage of the features of blockchain technology to increase the security of the ecosystem. Recently, as the existence of quantum computers has been felt, researchers have started to benefit from post-quantum cryptography to increase privacy and security. There has been an increase in data and asset protection in post-quantum blockchain-based solutions. To the best of our knowledge, there is no comprehensive review or taxonomy that provides a complete picture of post-quantum secure structures with privacy-preserving techniques that have the potential to be used in blockchain. This paper aims to close this gap by systematically examining these approaches and revealing the deficiencies in the existing literature and the development potential in these areas. The taxonomy examines the role of blockchain technology in post-quantum cryptography and emphasizes the potential of technologies such as zero-knowledge proof to ensure privacy in post-quantum blockchain-based systems. We also review the existing literature on addressing the performance overhead, interoperability, scalability, and security challenges in implementing post-quantum cryptography in zero-knowledge proof-enabled blockchain architectures that protect against quantum computing threats. The studies are collected from journal papers in widely used academic databases between 2018 and 2024. The studies are subjected to certain elimination criteria, and 13 studies are reviewed in detail. Our approach will facilitate discussions on future research directions by proposing the accessibility of post-quantum cryptography against quantum threats to blockchain systems and solutions to the challenges that arise in the integration phase.