Measuring the dispersion of photonic devices with small dispersion-length products is challenging due to the phase-sensitive, and alignment-intensive nature of conventional methods. In this letter, we demonstrate a quantum technique to extract the second-, and third-order chromatic dispersion of a short single-mode fiber using a fiber-based quantum nonlinear interferometer. The interferometer consists of two cascaded fiber-based biphoton sources, with each source acting as a nonlinear beamsplitter. A fiber under test is placed in between these two sources, and introduces a frequency-dependent phase that is imprinted upon the biphoton spectrum (interferogram) at the output of the interferometer. This interferogram contains within it the dispersion properties of the test fiber. Our technique has three novel features: (1) The broadband nature of the biphoton sources used in our setup allows accurate dispersion measurements on test devices with small dispersionlength products; (2) our all-fiber common-path interferometer requires no beam alignment or phase stabilization; (3) multiple phase-matching processes supported in our biphoton sources enables dispersion measurements at different wavelengths, which yields the third-order dispersion, achieved for the first time using a quantum optical technique.Chromatic dispersion is an important physical property that affects the propagation of optical pulses in photonic systems; in linear systems it is used for pulse shaping [1], while in nonlinear systems it plays an important role in soliton propagation, and affects the efficiency of many nonlinear interactions [2]. Dispersion characterization is then crucial for designing optimized photonic devices. Significant effort has been expended in past decades on extracting the chromatic dispersion of materials using classical light sources. Techniques such as time-of-flight [3], and modulation phase shift [4] were introduced to measure the dispersion of components with large dispersion-length products. Components with small dispersion-length products were characterized with temporal [5], and spectral [6] white-light interferometry (WLI), with the latter proving to be the more robust method against environmental noise [7].