Digital signatures guarantee the authorship of electronic communications. Currently used "classical" signature schemes rely on unproven computational assumptions for security, while quantum signatures rely only on the laws of quantum mechanics to sign a classical message. Previous quantum signature schemes have used unambiguous quantum measurements. Such measurements, however, sometimes give no result, reducing the efficiency of the protocol. Here, we instead use heterodyne detection, which always gives a result, although there is always some uncertainty. We experimentally demonstrate feasibility in a real environment by distributing signature states through a noisy 1.6 km free-space channel. Our results show that continuous-variable heterodyne detection improves the signature rate for this type of scheme and therefore represents an interesting direction in the search for practical quantum signature schemes. For transmission values ranging from 100% to 10%, but otherwise assuming an ideal implementation with no other imperfections, the signature length is shorter by a factor of 2 to 10. As compared with previous relevant experimental realizations, the signature length in this implementation is several orders of magnitude shorter. DOI: 10.1103/PhysRevLett.117.100503 Digital signatures [1] are ubiquitous in electronic communication, used in, for example, Email and digital banking. They guarantee the provenance, integrity, and transferability of messages. Currently used classical digital signature schemes, however, rely on unproven computational assumptions [2], and may become insecure especially if quantum computers can be built [3]. Quantum digital signatures (QDSs) [4][5][6][7][8][9][10], on the other hand, give information-theoretic security [7], loosely speaking based on the fact that nonorthogonal quantum states cannot be perfectly distinguished from each other.The first quantum signature schemes assumed tamperproof, "authenticated" quantum communication links. Intuitively, this could be accomplished using parameter estimation techniques similar to those used in quantum key distribution (QKD). How to achieve this was explicitly shown only recently [10,11]. In addition, recent quantum signature schemes [6,9], including our protocol, do not require long-term quantum memory. Importantly, this means that quantum signatures can be implemented with current technology, essentially similar to QKD setups. "Classical" signature schemes with information-theoretic security also exist [12][13][14], but rely on secret shared keys, which could be accomplished using QKD. Quantum signature schemes may have some advantages over schemes relying on shared keys generated using QKD. In particular, the quantum bit error threshold for a signature scheme is in practice less strict than for distilling a secret shared key [11]. In addition, the required postprocessing is less demanding. Exactly what signature schemes are the most efficient, however, remains an open problem.Note that most QDS protocols, including this one, use quantu...