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Based on a synthesis of three main ingredients: (i) the Shannon information in nonequilibrium systems, (ii) the semiclassical energy-time quantization rule, and (iii) the quasistatic information-energy correspondence, a new general rule for the quantization of quasistatic information states supported by an environment away from equilibrium is introduced if the history of the environment is known as a function of time in terms of its thermodynamic potential for information $T(t)ΔS(t)$ that is a free energy measuring the distance from equilibrium $ΔS(t)$, and $T(t)$ is the mean temperature of the environment at time $t$. This all new quasistatic information-time quantization rule is applied to the expanding universe using a phenomenological thermodynamic potential for information in the matter dominated era in order to find the eigen-informations of the persistent structures that are supported by the universe (or the local environments therein) at any given epoch, thus providing an information-theoretic foundation for formation of structures and rise of complexity with time that embodies the cosmic evolution as epitomized by the late Wheeler's famous conjecture ``{\it it from bit}". This theoretical procedure must also open new avenues for further research into the quantum theory of information and complexity in nonequilibrium thermodynamics.