Final Report Summary - QUACOCOS (Quantum Correlations in Complex Systems) The main objectives of the project “Quantum correlations in complex systems (QUACOCOS)” were the development of novel mathematical tools, capable of revealing and quantifying the paradigmatic quantum effect of entanglement and finding applications for these tools in complex quantum systems. The projects objectives were not only reached, but the results branched out and found unexpected applications and a wide impact. The first milestones in entanglement detection and quantification were already achieved in the early stages of the project and subsequently further refined and improved during the whole duration. The fellow developed tools that were able to detect entanglement in complex, i.e. multipartite and high dimensional systems, that were experimentally feasible and enabled also a quantification and classification of the entanglement structure. These results were published in a series of papers, two of them in the prestigious Physical Review Letters. The emerging classification showed intricate connections to entropy distributions of quantum marginal, a problem that was also pursued by the fellow and the scientist in charge and led to the first classification of rank distributions of multipartite quantum states, published in Linear Algebra and its Applications.Through a close collaboration with quantum optics experimentalists the developed criteria were tailored to specific experimental requirements and consequently used to reveal the largest dimensional entanglement achieved with photons so far. The results of these collaborations were published in the Proceedings of the National Academy of Sciences (PNAS) and Nature: communications and bear witness to the fact that also the applicability of the developed tools was successfully put to test. Having achieved the major goals of the proposal the fellow continued to pursue various research directions inspired from the resulting mathematical characterization of entanglement. He found novel applications of high-dimensional entanglement in device independent quantum key distribution and identified unexplored resource states for quantum algorithms called hyper-graph states.Finally the fellow turned to investigating the role of quantum entanglement in thermodynamics at the quantum scale. Having developed the right tools he was able to show that this effect manifests itself in various thermodynamical machines, such as the world’s smallest refrigerators, where it enhances cooling capacities, or in work storage and extraction capabilities. He worked out the thermodynamic resources required to generate correlations and entanglement in complex multipartite systems.While working on all of these projects the fellow collaborated with over 34 scientists from seven different European countries (Austria, Italy, Germany, Poland, Spain, Switzerland, UK) and extended collaborations also to India and the United States. His frequent visits to different research institutions and universities contributed tremendously to the career development of the fellow as well as the transfer of knowledge in Europe.The publications from this project are highly cited and the fellow has presented them in many conferences, workshops and seminars, ensuring the academic impact of the results.In summary the project was a thorough success in all areas and its results will continue to shape the research endeavours of the fellow for years to come.