Quantum computing research and development efforts have grown dramatically over the past decades, led in part by initiatives from governments around the world. Government quantum computing investments are often driven by national security or digital sovereignty concerns, with the language used depending on the geography involved. For example, a focus on “national security” and quantum computing is prominent in the USA, while European countries regularly focus on “digital sovereignty”. These phrases are often loosely defined and open to interpretation, and they share some common motivations and characteristics (but also have important differences). This paper identifies specific governmental entities typifying the national security/digital sovereignty perspectives, along with these organisations’ respective roles within national and international policy engagement in quantum computing. It analyses governmental structures, historical developments and cultural characteristics that contributed to this national security–digital sovereignty divide. Building on this analysis, we use the history of other technologies to illustrate how we might adapt tested policy approaches to modern political dynamics and to quantum computing specifically. We frame these policy approaches so that they do not overemphasise “digital sovereignty” or “national security”, but rather address interests shared across both concepts, with a view to facilitating international collaboration.