Puno to Arequipa
 

Oddly, I researched that same question and came up with a big zip, nada, nothin'. :unsure: Because there are multiple options in the reverse direction, I began to think that companies were simply driving their buses into Lake Titicaca rather than drive through those mountains again! ;) I never did learn the reason, but my research (which was not exhaustive) convinced me that for whatever reason, buses make a loop that includes the Arequipa to Puno stretch in only that one direction, A to P.

Good luck!

I'm not sure why you came up with zilch, kja

Check Cruz del Sur
https://www.cruzdelsur.com.pe/

4M Express has once daily service on that route, leaving quite early, including hotel pickup and dropoff (in downtown areas of Puno and Arequipa). IMO worth the extra charge to not deal with bus stations and transfers, if you don't mind the early time. It includes a few touristy stops. Quite a beautiful route.
Puno - Arequipa - 4M Store

You could also do a transfer to Chivay first, spend a night, and then continue to Arequipa the next day. Killawasi is nice, if you want a hotel in Colca Canyon (Yanque), easy to reach by taxi from central Chivay.

I would expect a hotel or travel agent in Puno or Arequipa (and maybe also Cusco) could also sell you that 4M ticket, not sure how much lead time you need to ensure hotel pickups though. The 4M buses have guides on board, helpful if you have altitude issues or need to be sure of getting to your hotel safely. The bus station in Puno is a place to be careful of luggage and rogue taxis.

We just went to Puno bus station and bought tickets for the next bus to Arequipa. There are several companies running the direct route. The problem is that only Cruz Del Sur operate during the day, the rest like Civa etc. Leave at silly o’clock and get in to Arequipa before dawn. Frankly, 4M, via chivay as suggested by Mlgb is the best bet. Check out rome2rio.com for times and operators.

The Cruz del Sur website says 3 daily departures, 8am, 3pm, and midnightish. Try not to take night buses. Scenery is too good and driving too bad. I don't know about other operators but would rather not use a random one if there are good options available with a little planning. Problem with relying on 3d party websites is that all departures may not be listed. I use Go2Peru for research only. Often there will be other departures not listed especially a continuation service.

Going through Chivay adds time and expense if it was not on the agenda. 4M does run a direct bus to AQP which would be a second best option if you did not want to include Chivay. It isn't much more than the others and worthwhile for the guide and transfers, as described.Provided you can handle the 6am pick up.