Quantifying uncertainty and error bounds is a key outstanding challenge in ocean state estimation and climate research. It is particularly difficult due to the large dimensionality of this nonlinear estimation problem and the number of uncertain variables involved. The "Estimating the Circulation and Climate of the Oceans" (ECCO) consortium has developed a scalable system for dynamically consistent estimation of global time-evolving ocean state by optimal combination of ocean general circulation model (GCM) with diverse ocean observations. The estimation system is based on the "adjoint method" solution of an unconstrained least-squares optimization problem formulated with the method of Lagrange multipliers for fitting the dynamical ocean model to observations. The dynamical consistency requirement of ocean state estimation necessitates this approach over sequential data assimilation and reanalysis smoothing techniques. In addition, it is computationally advantageous because calculation and storage of large covariance matrices is not required. However, this is also a drawback of the adjoint method, which lacks a native formalism for error propagation and quantification of assimilated uncertainty. The objective of this dissertation is to resolve that limitation by developing a feasible computational methodology for uncertainty analysis in dynamically consistent state estimation, applicable to the large dimensionality of global ocean models. Hessian (second derivative-based) methodology is developed for Uncertainty Quantification (UQ) in large-scale ocean state estimation, extending the gradient-based adjoint method to employ the second order geometry information of the model-data misfit function in a high-dimensional control space. Large error covariance matrices are evaluated by inverting the Hessian matrix with the developed scalable matrix-free numerical linear algebra algorithms. Hessian-vector product and Jacobian derivative codes of the MIT general circulation model (MITgcm) are generated by means of algorithmic differentiation (AD). Computational complexity of the Hessian code is reduced by tangent linear differentiation of the adjoint code, which preserves the speedup of adjoint checkpointing schemes in the second derivative calculation. A Lanczos algorithm is applied for extracting the leading rank eigenvectors and eigenvalues of the Hessian matrix. The eigenvectors represent the constrained uncertainty patterns. The inverse eigenvalues are the corresponding uncertainties. The dimensionality of UQ calculations is reduced by eliminating the uncertainty null-space unconstrained by the supplied observations. Inverse and forward uncertainty propagation schemes are designed for assimilating observation and control variable uncertainties, and for projecting these uncertainties onto oceanographic target quantities. Two versions of these schemes are developed: one evaluates reduction of prior uncertainties, while another does not require prior assumptions. The analysis of uncertainty propagation in the ocean model is time-resolving. It captures the dynamics of uncertainty evolution and reveals transient and stationary uncertainty regimes. The system is applied to quantifying uncertainties of Antarctic Circumpolar Current (ACC) transport in a global barotropic configuration of the MITgcm. The model is constrained by synthetic observations of sea surface height and velocities. The control space consists of two-dimensional maps of initial and boundary conditions and model parameters. The size of the Hessian matrix is 0(1010) elements, which would require 0(60GB) of uncompressed storage. It is demonstrated how the choice of observations and their geographic coverage determines the reduction in uncertainties of the estimated transport. The system also yields information on how well the control fields are constrained by the observations. The effects of controls uncertainty reduction due to decrease of diagonal covariance terms are compared to dynamical coupling of controls through off-diagonal covariance terms. The correlations of controls introduced by observation uncertainty assimilation are found to dominate the reduction of uncertainty of transport. An idealized analytical model of ACC guides a detailed time-resolving understanding of uncertainty dynamics. Keywords: Adjoint model uncertainty, sensitivity, posterior error reduction, reduced rank Hessian matrix, Automatic Differentiation, ocean state estimation, barotropic model, Drake Passage transport.