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Quantum digital signatures (QDSs) allow the sending of messages from one sender to multiple recipients, with the guarantee that messages cannot be forged or tampered with. Additionally, messages cannot be repudiated—if one recipient accepts a message, she is guaranteed that others will accept the same message as well. While messaging with these types of security guarantees are routinely performed in the modern digital world, current technologies only offer security under computational assumptions. QDSs, on the other hand, offer security guaranteed by quantum mechanics. All thus far proposed variants of QDSs require long-term, high quality quantum memory, making them unfeasible in the foreseeable future. Here, we present a QDS scheme where no quantum memory is required, which also needs just linear optics. This makes QDSs feasible with current technology.