  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.
