In emerging markets, the growth of supply chain networks and logistics industry leads to a significant increase in energy consumption and induces high carbon emissions. To design a sustainable low-carbon supply chain network, we consider a carbon emission-compliance green location-inventory problem. It is characterized by uncertain demand and volatile carbon prices under a multi-year emission regulation, inspired from the carbon-trading scheme in China. A two-stage stochastic mathematical model is built and is solved with a three-phase hierarchical metaheuristic on extensive numerical experiments, which mimic the business context of a supply chain network operated in China. The results show to which extent the carbon-trading emission-compliance scheme, with uncertainties in demand and carbon price, impacts the strategic decisions. Besides, carbon emissions and supply chain profits of the design solutions produced under alternative emission regulations are evaluated and discussed. We also underline the sensitivity of the amount of carbon emissions to the demand uncertainty, and to the level and volatility of the carbon price in the carbon-trading system. These results provide managerial insights for supply chain emitters, and indicate that reasonable and stable carbon prices should be maintained by governments in emerging markets.