This paper is motivated by the open questions concerning effective nonlinear state estimation (SE) approaches. The basic difficulty comes from the highly nonlinear functions relating measurements and voltages defined by the AC power flow models. Today's AC power flow SE approach is, therefore, a highly non-convex problem and, as such, it is prone to convergence problems and sub-optimal solutions. We describe how two recently proposed methods overcome this problem by following a common idea of mapping the voltage space into higher dimensional space in which the problem can be posed as a convex optimization problem because the measurements can be expressed as linear functions in a higher-dimensional space. It is intriguing that the semi-definite programming (SDP)-based SE approach and the direct non-iterative method-based SE approach both employ a similar mapping of voltages into higher dimensional matrix W of voltage products at neighboring buses as new states. These recently discovered similarities are described in some detail by posing both approaches without loss of generality on a three bus system. The two methods differ significantly in their approaches to re-computing actual voltages once the voltage products are estimated. The SDP-based SE approach utilizes the structure of the mapping and states a sufficient rank one condition for matrix W to ensure the unique reconstruction of the actual bus voltages. Both methods are computationally demanding. To overcome this inherent problem, an approximate distributed SDP-based SE algorithm is proposed by performing: 1) a decomposition of the large power grid networks into much smaller clusters where extensive information exchange is not needed; and 2) by performing a Lagrangian dual decomposition-based computation and message-passing within each cluster. The accuracy of the SDP-based distributed algorithm is illustrated by comparing the results to those obtained using the SDP-based SE estimator. Advantages of the SDP-based methods when compared to today's AC power flow based SE are illustrated by showing numerical problems experienced when using the IEEE test systems.