PaulRB:
Could you re-word that so it's a bit less self-contradictory?
I can't see your diagram. The forum guidelines so tell you how to attach a picture to your post.
Well, you are right, it is simultaneously - logic wise, because is one event that trigger consequential relay closing
Delta_G:
Looks like a shift register to me. I'm not sure how you think it could get any simpler unless you want to pull all 8 lines from the Arduino itself.
Simplest way would be to use 8 Arduino pins, as already mentioned. No extra chips need. But if you can't afford 8 pins, then either a shift register or maybe a decade counter of some kind. Those would need at least 2 Arduino pins.