We demonstrate a high-speed and high-resolution quantitative phase imaging method by implementing a synthetic aperture technique through using digital micromirror devices (DMDs). DMDs provide high speed steering of the illumination beam angle upon the sample that induces sample frequency shift, which is the basis of achieving high resolution in the quantitative phase imaging (QPI) system. With a high-speed camera for image acquisition, our QPI system achieves synthetic aperture imaging at >25 frame per second (fps). The high-speed imaging capability of the system allows for better observation of samples dynamics, especially in live biological structures, where motions could result in degraded imaging quality. In experiments, our synthetic aperture-based QPI system has resolved sub-diffraction limited structures of 220 nm periods and quantified red blood cell membrane fluctuations, which opens new avenues in material metrology and biological imaging applications.