If the idea of Eternalism as expressed by those like J.M.E. McTaggart and Sean Carroll is true (as much of physics seems to suggest), the idea of the present moment being more real than the past or future is just a biological illusion. If that's the case, then what does it mean for our consciousness to in some sense exist eternally at every moment in our lives? I have to imagine that we're continuously retreading the same path through space-time that is producing the same experience over and over, only experiencing it linearly one moment at a time because of the limitations of our perception. I see this as different from Nietzsche's idea of Eternal Recurrence; we aren't appearing in different incarnations that have identical experiences over infinite time but continually reliving the exact same experience.
Good explanation of Eternalism and Presentism here.