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Real-time control of a smart distribution grid with renewable energy based generators requires accurate state estimates, that are typically based on measurements aggregated from smart meters. However, the amount of data/measurements increases with the scale of the physical grid, posing a significant stress on both the communication infrastructure as well as data processing control centers. This paper first investigates the effect of geographical footprint of distributed generation (DG) on the voltage states of a smart distribution system. We demonstrate that the strong coupling in the physical power system results in estimated voltage phasors exhibiting a correlation structure that allows for compression of measurements. Specifically, by exploiting principles of 1D and 2D compressed sensing, we develop two approaches, an indirect and direct method for state estimation starting from compressed power measurements. We illustrate the effectiveness of voltage estimation with significantly low number of random spatial, temporal as well as spatio-temporal power measurements using the IEEE 34 node distribution test feeder and a larger 100 node radial distribution system. Results show similar performance for both methods at all levels of compression. It is observed that, even with only 50% compressed power measurements, both methods estimate the states of the test feeder with high level of accuracy.