Solid-state broadband light emitters in the visible have revolutionized today's lighting technology achieving compact footprints, flexible form factors, long lifetimes, and high energy saving, although their counterparts in the infrared are still in the development phase. To date, broadband emitters in the infrared have relied on phosphor-downconverted light emitters based on atomic optical transitions in transition metal or rare earth elements in the phosphor layer resulting in limited spectral bandwidths in the near-infrared and preventing their integration into electrically driven light-emitting diodes (LEDs). Herein, phosphor-converted LEDs based on engineered stacks of multi-bandgap colloidal quantum dots (CQDs) are reported as a novel class of broadband emitters covering a broad short-wave infrared (SWIR) spectrum from 1050-1650 nm with a full-width-half-maximum of 400 nm, delivering 14 mW of optical power with a quantum efficiency of 5.4% and power conversion efficiency of 13%. Leveraging the electrical conductivity of the CQD stacks, further, the first broadband SWIR-active LED is demonstrated, paving the way toward complementary metal-oxide-semiconductor integrated broadband emitters for on-chip spectrometers and low-cost volume manufacturing. SWIR spectroscopy is employed to illustrate the practical relevance of the emitters in food and material identification case studies.