The UNESCOS project explores new frontiers for condensed matter physics: the interplay between new states of matter and superconductivity in strongly correlated electron systems. Unlocking this fundamental issue will provide materials scientists with new insights on how to design and produce new superconductors operating at higher temperature. This line of research will ultimately lead to technological breakthrough, new and more efficient avenues to produce, store and transmit electricity. Dealing with the enigmatic “pseudogap” state out of which high temperature superconductivity emerges in the phase diagram of cuprate superconductors, UNESCOS focuses on the study of unconventional charge density instabilities and aim to develop the concept of unconventional superconductivity driven by quantum criticality. The UNESCOS project is a jointed research program involving physicists from LNCMI, LLB and IPhT, whose works have received an important visibility in the last few years, bringing new concepts to this field: (i) Fermi surface reconstruction and stabilization of a charge density wave order under magnetic field, (ii) observation of an intra-unit-cell magnetic order in the pseudogap state using polarized neutron scattering technique, (iii) condensations of new phases, potentially responsible for the pseudogap state, induced by antiferromagnetic quantum fluctuations. Taken separately these works may seem to promote different and apparently conflicting physical pictures. The UNESCOS project takes up the challenge to bridge together phenomena, previously supposed unrelated and promote the emergence of a unified theoretical picture within the framework of a new theory for the pseudogap state implying a multicomponent order parameter mixing a quadrupolar density wave order and d-wave superconductivity. This new, controlled and predictive theory will be developed and specific calculations will be performed to predict and explain new experimental observations that will be carried out within the project. Indeed the theoretical work will be performed in synergy with thermodynamic (sound velocity), diffraction and spectroscopy (neutron and X-ray) measurements providing key information on the microscopic nature and symmetry of the anomalous electronic phenomena. These experiments will require technologies that are available only in large facilities (Neutron source, Synchrotron, High-Magnetic-Field laboratory).