In a general aspect, a quantum streaming kernel processes a data stream. In some aspects, an input stream of data is converted to an output stream of data by repeatedly receiving new portions of the input stream; encoding each new portion into an internal quantum state of a quantum processor; measuring a first part of the internal quantum state while maintaining coherence of a second part of the internal quantum state; and producing the output stream of data based on the measurements. In some cases, a history of the input stream is preserved by the coherence of the internal quantum state, and the measurements contain information based on the history of the input stream.