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The second quantum revolution has been producing groundbreaking scientific and technological outputs since the early 2000s; however, the scientific literature on the impact of this revolution on the industry, specifically on start-ups, is limited. In this paper, we present a landscaping study with a gathered dataset of 441 companies from 42 countries that we identify as quantum start-ups, meaning that they mainly focus on quantum technologies (QT) as their primary priority business. We answer the following questions: (1) What are the temporal and geographical distributions of the quantum start-ups? (2) How can we categorize them, and how are these categories populated? (3) Are there any patterns that we can derive from empirical data on trends? We found that more than 92% of these companies have been founded within the last 10 years, and more than 50% of them are located in the US, the UK, and Canada. We categorized the QT start-ups into six fields: (i) complementary technologies, (ii) quantum computing (hardware), (iii) quantum computing (software/application/simulation), (iv) quantum cryptography/communication, (v) quantum sensing and metrology, and (vi) supporting companies, and analyzed the population of each field both for countries, and temporally. Finally, we argue that low levels of quantum start-up activity in a country might be an indicator of a national initiative to be adopted afterwards, which later sees both an increase in the number of start-ups, and a diversification of activity in different QT fields.