Many cryptographic techniques have been proposed to conceive a secure and privacy-oriented vehicular ad hoc network (VANET) for its practical deployment. The security of these techniques requires a common secret key to be shared between the communicating entities or depend upon the premise that some mathematical problems are computationally hard. However, because of the open nature of the wireless medium, the communication cannot be kept confidential and is prone to eavesdropping. Furthermore, with the arrival of quantum computers, these techniques are prone to quantum attacks—the time complexity of the assumed hard problem gets reduced from millions of years to a few seconds. In this paper, we propose a conditional privacy-preserving authentication scheme based on a quantum key distribution protocol for vehicle-to-infrastructure (V2I) communication. Our scheme inherits the properties of the quantum key distribution protocol. It does not require a secret authentication key to be transmitted conventionally and is resistant to quantum attacks. Apart from protecting VANETs against generic security threats, including node impersonation, message tampering, and repudiation, our scheme defends VANETs against man-in-the-middle attacks, replay attacks, etc. Besides, our protocol ensures message unlinkability, vehicle-identity privacy, and vehicle traceability if a vehicle misbehaves. The results obtained from the performance evaluation of our scheme confirm reasonable values of information leakage, key length, bit error probability, etc.