## 1 Introduction  
Digital signature has been prevalent in past decades and applied in lots of scenes, such as electronic payment, electronic commerce and electronic government affairs, with strict demands for security. To satisfy the special requirements for diverse application scenarios, many ramifications of classical signature have occurred. The most concerning issue is the security of the classical signature scheme, which depends on computational complexity of some intractable problems involving the factorization of large numbers and the discrete logarithm. However, these problems can be efficiently solved by quantum algorithms with the development of quantum computation. For example, the former can be solved in polynomial time by [PERSON]'s quantum prime factorization algorithm [1]. [PERSON]'s algorithm [2] poses a great threat to symmetric cryptography by designing a more optimized brute force attack. Consequently, the classical signature schemes based on computational complexity are seriously struck and become insecure. Motivated by the merits of quantum technology, many scholars converted their attention from classical signature into quantum signature, the security of which is guaranteed by quantum non-cloning theorem [3] and [PERSON]'s uncertainty principle [4]. The development of quantum signature mainly relies on two essential techniques involving quantum teleportation and quantum encryption algorithm. Recently, quantum walks have been developed as an approach to realize quantum teleportation protocols [5; 6], which can improve the efficiency of quantum teleportation in terms of entanglement generation and measurements. Afterwards, [PERSON] and [PERSON] [7] showed the experimental realization of state transfer based on quantum walks with two coins. [PERSON] et al. [8] investigated the experimental implementation of quantum teleportation using coined quantum walks. [PERSON] et al. [9] also proposed a quantum teleportation scheme for transmitting an arbitrary multi-qubit state via multi-walker quantum walks. In addition, in 2017, [PERSON] et al. [10] introduced the idea of developing quantum key distribution protocol using quantum walks. Inspired by the above-mentioned work, enthusiastic scholars have presented several achievements in quantum signature. For example, in 2019 [PERSON] et al. [11] presented a quantum blind signature scheme based on quantum walk-based cryptosystem. In the same year, [PERSON] et al. put forward an arbitrated quantum signature protocol with quantum walks on complete graphs [12] and on a closed cycle [13], in which the necessary entangled states do not need to be prepared in advance and they can be created naturally via quantum walks. Furthermore, when transmitting a \(d\)-dimensional quantum state, two projective measurements with \(d\) elements instead of one joint measurement with \(d^{2}\) elements are required. The projective measurements are much easier to implement than the joint measurement in real experiments [5; 6]. In 2020, [PERSON] et al. [14] suggested another arbitrated quantum signature protocol, where quantum walk-based teleportation is applied to transfer the encrypted message copy and boson sampling-based random unitary encryption is used to generate the signature. Quite recently, [PERSON] et al. [15] applied quantum walks into quantum blind signature and presented the corresponding quantum blind signature scheme. Furthermore, quantum walks have been demonstrated to be realizable in different physical systems [16; 17; 18] and real experiments [19; 20; 21]. This stimulates us to explore more possibility of quantum walks into other types of quantum signature. Quantum proxy signature is an important type or branch of quantum signature and its concept was first proposed by [PERSON] et al. [22] in 1996. In 2001, [PERSON] and [PERSON] [23] introduced quantum mechanics into digital signature and proposed a quantum digital signature scheme based on one-way function.