Information about the band structure of quantum materials provides us with its the most essential ingredients to harness their properties. The characterization of quantum materials has accordingly been spearheaded by angle resolved photoemission spectroscopy (ARPES) because of its direct access to their band structure. However, certain materials require conditions in which ARPES may be limited, such as strong electromagnetic fields, field-effect devices, or cryogenic conditions. The scanning tunneling microscope operates well in those environments and it could complement ARPES investigations through Fourier transform tunneling spectroscopy (FT-STS), also referred to as quasiparticle interference imaging. The STM obtains momentum space information by measuring the local density of states (LDOS) of the standing wave patterns that are created by the elastic scattering of electron waves at surface discontinuities, such as steps of point-defects. However, FT-STS is notoriously slow because it involves the measurement of the LDOS for many energies over a large field of view and it can occupy an STM for days for a single map. We demonstrate here our work to fundamentally speed-up FT-STS measurements by the introduction of two methods: compressive sensing and parallel spectroscopy. Compressive sensing allows us to measure fewer LDOS locations and parallel spectroscopy enables us to measure the LDOS faster than using conventional tunneling spectroscopy. In combination we achieve up to 1000-fold faster mapping times, reducing FT-STS investigations from days to hours. We discuss the fundamentals of compressive sensing and parallel spectroscopy and look at future applications of fast QPI.