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Concerning the logged time, for time tracking purpose, the time that you spend executing some Test Run is not logged automatically in Jira; you can manually log it on the Test Execution issue though, either from the Test Run execution screen using a logging shortcut/Timer or from the corresponding Test Execution issue using the standard time tracking mechanism from Jira.
Remember that no matter what, if logged, this time is always logged on the Test Execution issue and is available on the related "Time Spent" custom field. It's not mandatory to log time; use it, if you need it.
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