Blockchain is a type of Distributed Ledger Technology (DLT) that has been included in various types of fields due to its numerous benefits: transparency, efficiency, reduced costs, decentralization, and distributivity realized through public-key cryptography and hash functions. At the same time, the increased progress of quantum computers and quantum-based algorithms threatens the security of the classical cryptographic algorithms, in consequence, it represents a risk for the Blockchain technology itself. This paper briefly presents the most relevant algorithms and procedures that have contributed to the progress of quantum computing and the categories of post-quantum cryptosystems. We also included a description of the current quantum capabilities because their evolution directly influences the necessity of increasing post-quantum research. Further, the paper continues as a guide to understanding the fundamentals of blockchain technology, and the primitives that are currently used to ensure security. We provide an analysis of the most important cryptocurrencies according to their ranking by market capitalization (MC) in the context of quantum threats, and we end up with a review of post-quantum blockchain (PQB) schemes proposals.