  This work presents instructive, yet comprehensive derivation of quantized
gravity theories in relativistic, classical, and semi-classical spacetime
structure based on the Poincar\'e, Galilean, and Bargmann algebra,
respectively. The technique of algebra gauging to construct the spacetime
dynamics - inspired by the approach of notable previous works - is introduced
to complement the standard vielbein formulation. The key characteristics and
anomalies of Galilean gravity will then be analyzed: the degenerate metric
structure, the additional degree of freedom in metric connection and the
additional necessary conditions of Galilean invariance among others. General
metric connection solution in Galilean spacetime differs fundamentally from
that of general relativity; this will be thoroughly investigated and an
explicit formula for such solution - equivalent to the parameterization by
Hartong and Obers (2015) - shall be derived. Multiple derivations of the
Bargmann algebra will be provided, together with both physical and algebraic
motivation for the extended Bargmann frame bundle. Finally, the physical impact
of constraining temporal torsion in classical spacetime will be discussed with
emphasis on the geometrical interpretation of time foliations.
