“Steal the Staff of Arcane Authority” is a Miscellaneous Quest in The Elder Scrolls V: Skyrim. This can acquired from Malur Seloth.
Talk with Malur Seloth in the Jarl's Longhouse in Winterhold. Talk with him about his connection to the College of Winterhold. he will admit he does not have any. Still, to keep up appearances he will ask you to go and steal the Staff of Arcane Authority.
The Staff is found in The Frozen Heath, the Inn just across from the long house. Slip inside and wait until just after midnight. The person the staff belongs to, Nelacar, will go to sleep around midnight. He be asleep until 7:00 am. This is the best time to slip in and steal the staff. It is found underneath the counter that he has in his room. Sneak in while he is asleep, grab the Staff and slip back out.
Head back over to the Jarl's Longhouse and talk with Malur Seloth. Tell him you got the staff and he will give you the reward.