  Correlations between different partitions of quantum systems play a central
role in a variety of many-body quantum systems, and they have been studied
exhaustively in experimental and theoretical research. Here, we investigate
dynamical correlations in the time evolution of multiple parts of a composite
quantum system. A rigorous measure to quantify correlations in quantum dynamics
based on a full tomographic reconstruction of the quantum process has been
introduced recently [\'A. Rivas et al., New Journal of Physics, 17(6) 062001
(2015).]. In this work, we derive a lower bound for this correlation measure,
which does not require full knowledge of the quantum dynamics. Furthermore we
also extend the correlation measure to multipartite systems. We directly apply
the developed methods to a trapped ion quantum information processor to
experimentally characterize the correlations in quantum dynamics for two- and
four-qubit systems. The method proposed and demonstrated in this work is
scalable, platform-independent and applicable to other composite quantum
systems and quantum information processing architectures. We apply the method
to estimate spatial correlations in environmental noise processes, which are
crucial for the performance of quantum error correction procedures.
