The work can be divided into three fields of research. The first field of research is the theory of temporal quantum correlations. These are the probabilities that occur if a sequence of measurements is made on a single system, see also Fig. 1. An important task is to study the correlations that can arise in quantum mechanics. In [J. Hoffmann et al., New J. Phys. 20, 102001 (2018)] we first determined the polytope of temporal quantum correlations coming from the most general measurements. We then showed that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identified the correlations in the simplest scenario that can not be reached by two-dimensional systems. This leads to a temporal inequality for a dimension test. We worked with an experimental group on the implementation using trapped ions [C. Spee et al., New J. Phys. 22, 023028 (2020)]. Finally, we discussed in detail the dimension dependence of the temporal correlations, giving also constructions of nonlinear witnesses for the quantum dimension [Y. Mao et al., arXiv:2005.13964] and introduced the notion of genuine multipartite entanglement in time [S. Milz et al., SciPost Phys. 10, 141 (2021)].
The second field of research concerns the connection between temporal correlations and the quantumness of a system. One point is here the derivation of noncontextuality inequalities. We provided a general method to derive Bell inequalities (which are also noncontextuality inequalities) for hypergraph states, which can be described by a nonlocal stabilizer formalism [M. Gachechiladze et al., Phys. Rev. Lett. 116, 070401 (2016)]. The violation of these inequalities increases exponentially with the particle number, but is robust against particle loss. More recently, we provided a systematic method to generalize Bell inequalities to more parties [F. Bernards et al., Phys. Rev. Lett. 125, 200401 (2020)] and this method can be used to study temporal Leggett-Garg or contextuality inequalities. A second major result concerns the nature of contextuality. We were able to show that the Kochen-Specker set with 18 vectors from Cabello et al. [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] is the minimal set for any dimension, verifying a longstanding conjecture by Peres [Z.P. Xu et al., Phys. Rev. Lett. 124, 230401 (2020)].
The third research area concerns the applications of temporal correlations. In [T. Simnacher et al., Phys. Rev. A 99, 062319 (2019)] we presented a general method to characterize and test quantum memories based on their ability to preserve coherence. We introduced a quality measure for quantum memories and characterize it in detail for the qubit case. The measure can be estimated from sparse experimental data by looking at few temporal correlations only. In [M. Gachechiladze et al., Phys. Rev. A 99, 052304 (2019)] we introduced a deterministic scheme of universal measurement-based computation, using only Pauli measurements a hypergraph state. This scheme leads to a different notion of computational depth, in contrast to the usual cluster-state scheme.
In summary, the project was extremely successful, not only in the sense of reaching its original ambitions goals, but also in opening novel research directions. The results have been published in international high-impact journals (such as Nature Communications, Physical Review Letters, and Review of Modern Physics) and attracted the attention of the international scientific community. In addition, four workshops with totally about 190 participants have been organized. Finally, several of the team members received prizes, e.g. or for the best paper on quantum foundations.
