Health departments, hospitals and companies around the world are rolling out the next wave in Covid-19 testing, which looks in a person’s blood for signs of past infection.

The new tests promise to give public-health and hospital officials a better idea of how widely the virus has spread and who can safely treat patients and stop social distancing. But uncertainty about the accuracy of some of the tests and unknowns surrounding immunity to Covid-19—the respiratory disease caused by the novel coronavirus—could limit their…

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