Using quantum technologies (QTs) to solve problems related to climate change is a key goal for many physicists at the research and development stage. Recent research anticipates numerous real-world applications for quantum technologies that will address climate change and further sustainable development goals. However, currently there is no guiding framework for implementing responsible, sustainable innovation, or criteria for evaluating the sustainability of QTs. The goal of this article is to augment previous responsible innovation (RI) analysis of, and recommendations for, quantum (...) innovation by emphasizing sustainability as a key value. This article will also provide specific recommendations for developing sustainable QTs and criteria to assess the sustainability of QTs. With increases in funding for quantum innovation and the predicted operationality of many QTs in the coming decades, this is a key moment to discuss values and shape the quantum innovation trajectory. By using an RI approach with an added emphasis on sustainability, this article offers tools for developing responsible, sustainable QTs that are sensitive to the climate change context. (shrink)