So what color management options have you set in Premiere?
And as you are comparing to QuickTime player, quite likely, you're on a Mac ... where the Mac ColorSync CM utility uses the bizarre Apple-only thing of applying the camera transform function of essentially gamma 1.96 to the display of Rec.709 video files, instead of the specified Rec.709 display transform function of essentially gamma 2.4.
You could try changing the viewer gamma setting in Premiere's CM to 1.96/QuickTime. Then inside Premiere on the Program monitor and outside on Macs without reference modes in QuickTime player, Chrome and Safari browsers, it will look very similar.
Of course ... on all Macs with Reference modes set to HDTV, which uses the proper display gamma, and nearly all other systems, including all broadcast/streaming spec systems, most PCs/Androids/TVs ... that will now look pretty dark.