Modern quantum engineering techniques enabled successful foundational tests of quantum mechanics. Yet, the universal validity of quantum postulates is an open question. Here we propose a new theoretical framework of Q-data tests, which recognises the established validity of quantum theory, but allows for more general -'post-quantum' -scenarios in certain physical regimes. It can accommodate both models with modified wave dynamics and correlations beyond entanglement. We discuss its experimental implementation suited to probe the nature of strong nuclear interactions. In contrast to the present accelerator experiments, it shifts the focus from high-luminosity beam physics to individual particle coherent control. A successful implementation of the proposed scheme would not only provide insight into the fundamental physics but also establish new devices for quantum information processing operating at unprecedented scales.