  How the macroscopic quantum coherence can resist to the decoherence attacks
of high temperature is a major challenge for the science of the 21st century.
Superstripes 2017 conference held in Ischia on June 2017 has been focused on
the new physics of high $T_c$ superconductors made of complex quantum matter.
Today the standard model of high $T_c$ superconductivity which grabs the
physics of complex quantum matter is the multi-band Hubbard model where the
dome of $T_c$ occurs by driving the chemical potential in the proximity of a
topological Lifshitz transition. The multi-gap superconductivity in the $T_c$
dome is driven by exchange interaction between a condensate in the BEC-BCS
crossover which coexists with second BCS condensates. The proximity to Lifshitz
transitions in correlated electronic systems gives the ubiquitous arrested
phase separation observed in all high temperature superconductors. Non
Euclidean filamentary hyperbolic geometry is needed for the space description
of superstripes textures produced by the coexistence of short range CDW
puddles, hole poor SDW puddles and self organized dopants rich puddles. A road
map to room temperature superconductors in particular organic compounds made of
superlattices of quantum wires driven by Fano resonances with one of the
condensates in the BEC-BCS crossover has been proposed.
