While artificial intelligence has been attracting increasing investment and controversy, a new, and potentially much more powerful technology, Quantum Computing, is on the business horizon. Already, organizations are assessing its potential for addressing a range of computationally challenging issues, such as last-mile optimization and cryptanalysis to break encryption algorithms, and are exploring possible applications in a range of business areas. While the advent of Quantum Computing may bring an array of opportunities and benefits, the companies developing and applying Quantum Computing technologies will face, and have to address, new sets of corporate social, ethical, economic, technical and environmental responsibilities. Using a qualitative research method based on secondary sources, this article examines the implications for Quantum Computing for corporate digital responsibility, and assesses the guidance currently being offered by leading consultancies for organisations deploying Quantum Computing. The results suggest that although there is some guidance on how companies can apply and exploit Quantum Computing technology, there is as yet little consideration of the possible downsides, including, for example, potential negative environmental impacts. Further research is needed to develop a more balanced assessment of benefits and disbenefits as Quantum Computing is deployed in the corporate world and wider global society.