Blockchain is an expanding technologythat offers benefits when applied to supplychain management. This distributed ledgertechnology is combined with supply chainsfor its intrinsic characteristics. For instance,traceability, immutability, and more arediscussed. In this paper, we first presentprivacy-related challenges encountered whenapplying blockchain technology to this work.These include dealing with immutable data,providing the anonymity of end-users, as wellas their accountability. The risks associatedwith the leakage of medical data show thenecessity of protecting patient privacy. Forthis sake, we delve into blood donation,clinical trial, PPE tracking supply chainsand highlight their privacy requirements.We introduce anonymous signatures, mixingservices, and other cryptographic techniques,which satisfy these requirements throughlocal anonymity, unlikability, but alsoaccountability in specific contexts.