A group signature scheme enables members of a group to sign messages on behalf of the group in a manner that conceals the identity of the individual signer. The identification of the member responsible for a particular signature is exclusively within the authority of a designated group manager. Existing state-of-the-art group signatures are based on number-theoretic hardness assumptions such as the integer factorization problem and the discrete logarithm problem. These protocols are not secure in the presence of quantum computers because quantum algorithms like Shor’s algorithm can be utilized to undermine the security of these schemes. In this paper, we present a quantum cryptography-based group signature (namely qGroup) protocol. Our design achieves all the security properties of a group signature. In addition, it is secure against attacks by quantum adversaries. qGroup performs better than the existing quantum cryptography-based group signature in the literature as qGroup employs only single qubits, simple measurement operators, and single-qubit quantum gates. We also present an application of qGroup to Vehicular Ad Hoc Networks (VANETs).