Recent results in compressive sampling have shown that sparse signals can be recovered from a small number of random measurements. This property raises the question of whether random measurements can provide an efficient representation of sparse signals in an information-theoretic sense. Through both theoretical and experimental results, we show that encoding a sparse signal through simple scalar quantization of random measurements incurs a significant penalty relative to direct or adaptive encoding of the sparse signal. Information theory provides alternative quantization strategies, but they come at the cost of much greater estimation complexity.