Abstract: The characterisation of strongly-correlated effects in quantum impurity systems (QIS) is particularly challenging due to the infinite size of the environment and the buildup of long-ranged entanglement in the system. Moving beyond standard (perturbative) transport analysis, we develop a framework based on the Numerical Renormalisation Group (NRG) and Variational Matrix Product States (VMPS) methods to resolve strong correlations in QIS. In our analysis, we harness the impurity's reduced density matrix' elements as witnesses for the formation of intra-impurity and impurity-environment strong correlations. We apply our scheme to several mesoscopic devices, including semiconductor quantum dots coupled to structured environments, as well as graphene islands. In my talk, I will focus on the example of a double-dot device, where the dots are coupled via off-resonant ballistic whispering gallery modes, where I find that (i) Kondo hybridization can win and form on each dot individually, or (ii) the whispering gallery modes can win and mediate between the two dots either a singlet or a novel nonlocal Kondo-like effect phase. To relate our findings to transport experiments, we extend the Chebyshev-MPS method to calculate the spectral functions of multi-orbit impurity systems. Extending our framework to finite temperature or out-of-equilibrium scenarios, we could develop a quantum impurity solver for DMFT applications. Our results advance our ability to understand and describe condensed matter physics in a plethora of systems.