Narratives shape public perceptions and policymaking around emerging technologies like quantum technologies (QTs), yet what narratives develop across different societal domains remains underexplored. This study analyzes narratives about QTs in 36 government documents, 163 business reports, and 2023 media articles published over the past 23 years, using a mixed-methods approach that combines topic modeling with qualitative thematic analysis. We find that the dystopian or utopian extremes associated with technologies such as artificial intelligence are largely absent from discourse about QTs. Media coverage tends to cover a broad range of topics, while business and government narratives emphasize technical milestones, economic competitiveness, and national security, frequently at the expense of questions about ethics, equity, and accessibility. We discuss the implications of this focus, particularly the risk that an emphasis on zero-sum geopolitical competition could foster a more closed and fragmented innovation ecosystem.