Security and privacy have always been key concerns for individuals in various edge-assisted services. In this paper, we present a feasible quantum solution to an important primitive of secure multiparty computations, i.e., Secure Multiparty Logical AND (SMLA), in which n participants can securely compute logical AND of n private bits. In order to ensure perfect security and achieve good feasibility, we introduce a semi-honest edge server and two non-collusive fog nodes, and design a secure and feasible edge-assisted quantum protocol for SMLA, which cleverly utilizes Secure Multiparty XOR to implement SMLA group by group. Furthermore, we focus on applications of this quantum primitive protocol and design two quantum protocols for Multiple Private Set Intersection and Anonymous One-vote Veto. Compared with classical related protocols, our proposed quantum protocols obtain higher security, which can be guaranteed by the basic principles of quantum mechanics.