Digital signature is a key security technology for authenticating systems and devices, thus enabling the existence of wide collaborative environments. This is also true for safetycritical systems that are constrained by strict performance requirements. Such applications are usually implemented through Multi-processors System-on-Chip (MPSoC). The dawn of quantum computing represents a threat for current cryptography, including the digital signatures. In order to prepare for such an event, electronic systems must integrate quantum-secure (postquantum) cryptography. Dilithium is one of the main alternatives for practical implementation of post-quantum signatures. While most of the attention has been given to the security analysis and single-core software implementation, the Dilithium MPSoC exploration for high performance has been neglected. To this end, this work presents two contributions. First, the design and exploration of optimized Dilithium multi-core implementations. Second, the deployment of Dilithium on real life MPSoCs used in automotive applications and operated with a commercial RTOS. Results show that Dilithium can be efficiently implemented and optimized on a multicore architecture, improving the performance up to 48% for key generation, 34% for signature and 42% for verification when compared to single core solutions.