In this thesis we summarize how canonical effective methods can be applied to describe quantum corrected gauge symmetries, non-adiabatic corrections in inflation and backreactions that seed correlations and inhomogeneities in field theories. Special attention will be paid to canonical structure and dynamics of quantum fluctuations (which can be interpreted as moments of a quantum state). We will use canonical methods to highlight the role of constraints in the quantum corrected low energy effective theory of gravity. Deformed covariance is shown to be a generic feature. Resolutions to seemingly different effective Hamiltonians are also discussed in this context. We then explore the effect of dynamical quantum fluctuations on effective potentials in inflation. We show how these fluctuations affect observables and help us characterize the quantum state, thus establishing a state-to-observable correspondence. Finally, we discuss the role of backreactions on background correlations and inhomogeneity growth. We find that distinct oscillation behaviors arise when correlations are present. Furthermore, trickling of infrared inhomogeneities to smaller length scales are demonstrated to exist with the inclusion of non-local effects from quantum moments.