Use a QWidgetAction and QCheckBox for a "checkable action" which doesn't cause the menu to close.
QCheckBox *checkBox = new QCheckBox(menu);
QWidgetAction *checkableAction = new QWidgetAction(menu);
checkableAction->setDefaultWidget(checkBox);
menu->addAction(checkableAction);
In some styles, this won't appear exactly the same as a checkable action. For example, for the Plastique style, the check box needs to be indented a bit.
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