This work introduces XMSS-T, a new stateful hash-based signature scheme with tight security. Previous hash-based signatures are facing a loss of security, linear in performance parameters such as the total tree height. Our new scheme can achieve the same security level but using hash functions with a smaller output length, which immediately leads to a smaller signature size. The same techniques also apply directly to the recent stateless hash-based signature scheme SPHINCS (Eurocrypt 2015), and the signature size is reduced as well. Being a little more specific and technical, the tight security stems from new multi-target notions of hash-function properties which we define and analyze. We show precise complexity for breaking these security properties under both classical and quantum generic attacks, thus establishing a reliable estimate for the quantum security of XMSS-T. Especially, we prove quantum query complexity tailored for cryptographic applications, which overcome some limitations of standard techniques in quantum query complexity such as they usually only consider worst-case complexity. Our proof techniques may be useful elsewhere. We also implement XMSS-T and compare its performance to that of XMSS (PQCrypto 2011), the most recent stateful hash-based signature scheme before our work.