Web 3.0 pursues the establishment of decentralized ecosystems based on blockchain technologies to drive the digital transformation of physical commerce and governance. Through consensus algorithms and smart contracts in blockchain, which are based on cryptography technologies, digital identity, digital asset management, decentralized autonomous organization, and decentralized finance are realized for secure and transparent digital economy services in Web 3.0 for promoting the integration of digital and physical economies. With the rapid realization of quantum devices, Web 3.0 is being developed in parallel with the deployment of quantum cloud computing and quantum Internet. In this regard, quantum computing first disrupts the original cryptographic systems that protect data security while reshaping modern cryptography with the advantages of quantum computing and communication. Therefore, this survey provides a comprehensive overview of blockchain-based Web 3.0 and its quantum and post-quantum enhancement from the ambilateral perspective. On the one hand, some post-quantum migration methods, and anti-quantum signatures offer potential ways to achieve unforgeable security under quantum attack for the internal technologies of blockchain. On the other hand, some quantum/post-quantum encryption and verification algorithms improve the external performance of the blockchain, enabling a decentralized, valuable, secure blockchain system. Finally, we discuss the future directions toward developing a provable secure decentralized digital ecosystem.CCS Concepts: • Security and privacy → Cryptography; • Theory of computation → Models of computation .