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The emergence of Variants of Concern (VOCs) of SARS-CoV-2 with increased transmissibility, immune evasion properties, and virulence poses a great challenge to public health. Despite unprecedented efforts to increase genomic surveillance, fundamental facts about the evolutionary origins of VOCs remain largely unknown. One major uncertainty is whether the VOCs evolved during transmission chains of many acute infections or during long-term infections within single individuals. We test the consistency of these two possible paths with the observed dynamics, focusing on the clustered emergence of the first three VOCs, Alpha, Beta, and Gamma, in late 2020, following a period of relative evolutionary stasis. We consider a range of possible fitness landscapes, in which the VOC phenotypes could be the result of single mutations, multiple mutations that each contribute additively to increasing viral fitness, or epistatic interactions among multiple mutations that do not individually increase viral fitness—a “fitness plateau”. Our results suggest that the timing and dynamics of the VOC emergence, together with the observed number of mutations in VOC lineages, are in best agreement with the VOC phenotype requiring multiple mutations and VOCs having evolved within single individuals with long-term infections.