According to English fencing manuals from the 15th and 16th century the quarterstaff is one of best hand-to-hand weapons one can use. The techniques used with it are mirrored or often described in the same section as polearms, so including it with polearms is historically accurate.Show me how you use a quarterstaff one handed, (it is RAW somehow, i know) I don't know who at wizards came up with the idea to make a simple stick one of the most powerful non magic weapons in the game.
I houserule quarterstaff to do 1d6 and be two handed use only. And it is a traditional monk weapon of course. The polearm mastery might apply to that one because a feat is a heavy cost but otoh you use both ends of a quartestaff by default, so if you keep the d8 as in RAW this is counted in imho.
This really should be changed by errata, to avoid handed duell style quarterstaff polearm master with shield rulius magistrarius build and such.
As for wielding, it is held in two hands at low guard and is in general a two-handed weapon, although some of the attacks and parries are done with one hand letting go with either the leading or trailing hand.
Finally a real quarterstaff is not a stick, it is hardwoord and it is milled from the trunk of the tree not a branch (the "quarter" in quarterstaff refers to how it is sawed).