In measurement-based quantum computation (MBQC), the process of quantum computation is driven by local measurements, applied to a suitable entangled quantum state. This quantum state thereby becomes a resource for this computational scheme.

Cluster states and AKLT states of spin 3-2 or 2 are examples of computationally universal such resources. Further, it has been shown that, in the presence of symmetry, such states are surrounded by entire physical phases (in the condensed matter sense) of other quantum states with the same computational capabilities. Such phases are called computational phases of quantum matter.

I give an introduction into the history of the subject, and what has been learned and achieved so far. I emphasize the existence of computationally universal phases of quantum matter in 2D, both in SPT- and SET-ordered systems.

I conclude with an experiment (arXiv:2601.03426) that recently was conducted on an IBM quantum computer, and which verified four essential predictions for computational phases of matter.