Quantum-resistant authentication becomes crucial for securing vehicular ad-hoc networks (VANETs). However, existing solutions fail to support pseudonym unlinkability, and lacks a mechanism to ensure public verification of tracing and avoid malicious accusations against innocent vehicles for ensuring tracing robustness. In this paper, we propose a lattice-based robust and conditional privacy-preserving authentication (LRCPA) scheme for VANETs, which enables anonymous authentication on vehicles and adopts the small integer solution problem to withstand quantum attacks. Specially, obfuscated expiration dates are set for vehicles' credentials and pseudonyms to guarantee the unlinkability of new and old pseudonyms. The public verification of tracing is ensured by adopting vehicles' commitments or signatures on their real identities and pseudonyms as the proof of tracing, and tracing robustness is provided by employing vehicles' credentials issued by a central authority as the proof of malicious accusations or framing behaviors by the corrupted authority during tracing. LRCPA not only achieves message authentication and integrity, non-repudiation, anonymity, conditional privacy, unlinkability, key escrow freedom, public verification of tracing, and tracing robustness, but also effectively withstands most known attacks. Comprehensive performance analysis and evaluation show the potential usability of LRCPA.