  We elaborate on quantum geometric information flows, QGIFs, and emergent
(modified) Einstein-Maxwell and Kaluza-Klein, KK, theories formulated in
Lagrange-Hamilton and general covariant variables. There are considered
nonholonomic deformations of Grigory Perelman's F- and W-functionals
(originally postulated for Riemannian metrics) for describing relativistic
geometric flows, gravity and matter field interactions, and associated
statistical thermodynamic systems. We argue that the concept of Perelman
W-entropy presents more general and alternative possibilities to characterize
geometric flow evolution, GIF, and gravity models than the Bekenstein-Hawking
and another area-holographic type entropies. Formulating the theory of QGIFs, a
set of fundamental geometric, probability, and quantum concepts, and methods of
computation, are reconsidered for curved spacetime and (relativistic) phase
spaces. Such generalized metric-affine spaces are modeled as nonholonomic
Lorentz manifolds, (co) tangent Lorentz bundles, and associated vector bundles.
Using geometric and entropic and thermodynamic values, we define QGIF versions
of the von Neumann entropy, relative and conditional entropy, mutual
information, etc. There are analyzed certain important inequalities and
possible applications of G. Perelman and related entanglement and R\'{e}nyi
entropies to theories of KK QGIFs and emergent gravitational and
electromagnetic interactions. New classes of exact cosmological solutions for
GIFs and respective quasiperiodic evolution scenarios are elaborated. We show
how classical and quantum thermodynamic values can be computed for cosmological
quasiperiodic solutions and speculate how such constructions can be used for
explaining structure formation in dark energy and dark matter physics.
