Compressed sensing or compressive sampling (CS) is a signal processing technique for efficiently acquiring and reconstructing sparse signals by solving underdetermined linear systems. In practice, CS needs to be accompanied by a quantization process. That is, after sampling the signals, we represent the measurements using discrete data, e.g. 0s and 1s, and recover the signals from the quantized measurements. In this talk, I will discuss how to extend the noise-shaping quantization methods beyond the case of Gaussian measurements to structured random measurements, including random partial Fourier and random partial Circulant measurements. This is joint work with Rayan Saab