Localized communication between vehicles and their surrounding environment (V2X) is a key technology to enable Cooperative Intelligent Transportation Systems (C-ITS) aiming at road safety, traffic flow and driving comfort. Security services based on Elliptic Curve Cryptography (ECC) for authenticity and confidentiality (mostly application-dependent) have been chosen to meet the hard constraints of low latency safety communications and limited bandwidth radio communication in dense traffic conditions. Due to threats raised by Quantum Computers (QC), the classical asymmetric cryptographic algorithms could be broken impacting the Public Key Infrastructure (PKI)-based security solutions, with negative safety consequences on the (semi)-autonomous vehicles and road users. Our project (TAM: Trusted Autonomous Mobility) [18] is focusing on end-to-end cybersecurity and privacy for innovative services in the field of cooperative, connected and automated mobility (CCAM). One main objective is to find suitable quantum safe schemes to replace the current cryptographic standards based on ECC which are used in V2X communications. After defining the main requirements and key performance indicators for C-ITS, a benchmarking of current NIST pre-standards PQC algorithms was performed to assess the feasibility and performances in C-ITS applications and based on the results a best fit solution is selected.