Learning from correlations: what parts of quantum states tell about the whole

The aim of this thesis is to deepen the understanding of correlations between particles in quantum mechanical systems. It focuses on finding relationships between the correlations among different parts of the system, as well as revealing their limitations in multi-qubit states.
In particular, new answers to the quantum marginal problem are found, i.e., the question of whether knowledge of the subsystems of certain particle numbers allows fixing a global quantum state uniquely. Among other things, it is shown that in many cases, certain sets of two-particle reduced states determine a joint four-particle state uniquely.
Furthermore, it is shown that the set of correlations in multi-qubit states naturally decomposes into an odd and an even component, where often one component uniquely fixes the other. This finding is consequently applied to the problem of entanglement detection and the characterization of ground states of Hamiltonians.
In the second part of the thesis, interrelations between correlation quantifiers of degree two, known as Sector lengths, are established and connected to quantum mechanical properties of states. It is shown that Sector lengths are helpful for the detection of entanglement, and that they are subject to monogamy-like constraints, limiting the amount of concurrent correlations between different particles.
Consequently, it is investigated which additional information for the task of entanglement detection is yielded by higher-order invariants, in particular higher moments of the distribution of correlation measurements.
The third part considers the problem of entanglement detection in experimentally limited scenarios: It analyses the capabilities of entanglement detection having access to expectation values of two product observables only. In systems of restricted dimensionality, necessary and sufficient criteria to be useful for such tasks are developed for pairs of such observables.
The last chapter extends the scope of the thesis via a theoretical assessment of quantum memories. As important building blocks for future applications of quantum mechanics, like quantum computers and quantum communication, these memory devices need to store quantum states faithfully for the corresponding task. In order to being able to characterize this property of quantum memories sufficiently, abstract criteria for memory performance measures are developed. Consequently, three such measures based on the coherence of quantum states are defined and their properties are determined.