What is the secret human characteristic the Arcturians so desire to understand and utilize? We have been told their Grey robots, possessing artificial intelligence, seek to understand our emotions and immortality, allowing us to reincarnate again and again. Are those actually Arcturian goals as well?

You are correct here, that there is more to the story than the alien Grey robotic beings simply wanting to be more like the flesh and blood humanoid races they encounter, because they see the enjoyment the soul-based physical beings have in exercising their emotions, their passions, through excitement if not lofty intentions, at least on the part of extraterrestrials. They are fascinated with sexual attraction as they see it is such a powerful force when such emotions are foreign to them entirely. So that is another human attribute they are highly curious about. The fact we reincarnate is another item on their wish list because they are aware enough of their circumstances to know they are different than people and extraterrestrial physical beings. The Arcturians are not aware of their own reincarnation and so that is a shared desire of both Arcturians and their robotic alien Grey worker bees to solve the riddle of how humans can persist through a kind of recycling into spirit form and then back into the physical again and again. They also know there is something unique about human consciousness that has a kind of power they lack, and this is extremely upsetting for them, being so convinced of their vast superiority, but it is solely based on their technological prowess and their high level of intellectual ability compared to the average human being. So there are several things intriguing to the Arcturians about the human makeup, and the genetic program of the alien Greys is serving them as well in the pursuit of deeper knowledge and insight about the human makeup.