Promoters of blockchain-based systems such as cryptocurrencies have often advocated for the anonymity these provide as a pledge of privacy protection, and blockchains have consequently been envisioned as a way to safely and securely store data. Unfortunately, the decentralized, fully-replicated and unalterable nature of the blockchain clashes with both French and European legal requirements on the storage of personal data, on several aspects such as the right of rectification and the preservation of consent. PriCLeSS aims to establish a cross-disciplinary partnership between Computer Science and Law researchers to understand and address the legal and technical challenges associated with data storage in a blockchain context.