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We elaborate on quantum geometric information flows, QGIFs, and emergent(modified) Einstein-Maxwell and Kaluza-Klein, KK, theories formulated inLagrange-Hamilton and general covariant variables. There are considerednonholonomic deformations of Grigory Perelman's F- and W-functionals(originally postulated for Riemannian metrics) for describing relativisticgeometric flows, gravity and matter field interactions, and associatedstatistical thermodynamic systems. We argue that the concept of PerelmanW-entropy presents more general and alternative possibilities to characterizegeometric flow evolution, GIF, and gravity models than the Bekenstein-Hawkingand another area-holographic type entropies. Formulating the theory of QGIFs, aset of fundamental geometric, probability, and quantum concepts, and methods ofcomputation, are reconsidered for curved spacetime and (relativistic) phasespaces. Such generalized metric-affine spaces are modeled as nonholonomicLorentz manifolds, (co) tangent Lorentz bundles, and associated vector bundles.Using geometric and entropic and thermodynamic values, we define QGIF versionsof the von Neumann entropy, relative and conditional entropy, mutualinformation, etc. There are analyzed certain important inequalities andpossible applications of G. Perelman and related entanglement and R\'{e}nyientropies to theories of KK QGIFs and emergent gravitational andelectromagnetic interactions. New classes of exact cosmological solutions forGIFs and respective quasiperiodic evolution scenarios are elaborated. We showhow classical and quantum thermodynamic values can be computed for cosmologicalquasiperiodic solutions and speculate how such constructions can be used forexplaining structure formation in dark energy and dark matter physics.

Kaluza--Klein gravity & cosmology emerging from G. Perelman's entropy functionals and quantum geometric information flows