After the spread of COVID-19 out of China, the evolution of the pandemic has shown remarkable similarities and differences between countries around the world. Eventually, such characteristics are also observed between different regions of the same country. Herewith, we introduce a general method that allows us to compare the evolution of the pandemic in different localities inside a large territorial country: in the case of the present study, Brazil. To evaluate our method, we study the heterogeneous spreading of the COVID-19 outbreak until May 30th, 2020, in Brazil and its 27 federative units, which has been seen as the current epicenter of the pandemic in South America. Each one of the federative units may be considered a cluster of interacting people with similar habits and distributed to a highly heterogeneous demographic density over the entire country. Our first set of results regarding the time-series analysis shows that: (i) a power-law growth of the cumulative number of infected people is observed for federative units of the five regions of Brazil; and (ii) the distance correlation calculated between the time series of the most affected federative units and the curve that describes the evolution of the pandemic in Brazil remains about 1 over most of the time, while such quantity calculated for the federative units with a low incidence of newly infected people remains about 0.95. In the second set of results, we focus on the heterogeneous distribution of the confirmed cases and deaths. By applying the epidemiological susceptible-infected-recovered-dead model we estimated the effective reproduction number (ERN) [Formula: see text] during the pandemic evolution and found that: (i) the mean value of [Formula: see text] for the eight most affected federative units in Brazil is about 2; (ii) the current value of [Formula: see text] for Brazil is greater than 1, which indicates that the epidemic peak is far off; and (iii) Ceará was the only federative unit for which the current [Formula: see text]. Based on these findings, we projected the effects of increase or decrease of the ERN and concluded that if the value of [Formula: see text] increases 20%, not only the peak might grow at least 40% but also its occurrence might be anticipated, which hastens the collapse of the public health-care system. In all cases, keeping the ERN 20% below the current value can save thousands of people in the long term.