Blockchain and other Distributed Ledger Technologies (DLTs) have evolved significantly in the last years and their use has been suggested for numerous applications due to their ability to provide transparency, redundancy and accountability. In the case of blockchain, such characteristics are provided through public-key cryptography and hash functions. However, the fast progress of quantum computing has opened the possibility of performing attacks based on Grover's and Shor's algorithms in the near future. Such algorithms threaten both public-key cryptography and hash functions, forcing to redesign blockchains to make use of cryptosystems that withstand quantum attacks, thus creating which are known as post-quantum, quantum-proof, quantum-safe or quantum-resistant cryptosystems. For such a purpose, this article first studies current state of the art on post-quantum cryptosystems and how they can be applied to blockchains and DLTs. Moreover, the most relevant post-quantum blockchain systems are studied, as well as their main challenges. Furthermore, extensive comparisons are provided on the characteristics and performance of the most promising post-quantum public-key encryption and digital signature schemes for blockchains. Thus, this article seeks to provide a broad view and useful guidelines on post-quantum blockchain security to future blockchain researchers and developers.