Born in New York, Hauptman graduated from Columbia in 1939. After two years spent with the US Census he was drafted into the US Air Force where he spent much of the war as a radar instructor. In 1947 he joined the staff of the Naval Research Laboratory, Washington DC. At the same time he pursued his doctorate (awarded 1952) at the University of Maryland. In 1970 Hauptman moved to the Medical Foundation, Buffalo, as its research director, becoming president in 1988.

In collaboration with his Washington colleague Jerome Karle, Hauptman has made a significant advance in the use of x-ray crystallography. The classic x-ray photograph of a crystal has a diffraction pattern of dots and it was the task of the crystallographer to find a molecular structure that would give this pattern. A likely structure would be assumed and tested against the x-ray data. The method worked reasonably well when restricted to simple structures.

Further progress was made when crystallographers worked out how to apply Fourier analysis to the scattered x-rays. The difficulty was that, although the amplitudes of the waves could be deduced, their phase could not be determined in this way. In 1953, Hauptman and Karle showed mathematically how the phase problem could be overcome. Their method, known as the direct method, was mainly statistical and allowed structures to be determined much more quickly. Whereas previously a 15-atom molecule might require many months' work, the structure of larger molecules, following the work of Hauptman and Karle, could be worked out in a matter of days.

For his work in this area Hauptman shared the 1985 Nobel Prize for chemistry with Karle.