The Polearm Master feat (PHB, p. 168) says:
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
The War Caster feat (PHB, p. 170) says:
When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature.
On this very topic, Mearls has said it requires an attack with the same weapon that's granting you the trigger:
Q: Do Polearm Master and War Caster combine to allow a magic user to make a spell opportunity attack when they enter reach?
A: No – polearm master applies only if you use the weapons it lists to make the attack
Another question on this very topic: Which is likely to provide more chances to opportunity attack, Polearm Master or Sentinel (with War Caster)?
But since then, we now have spells like booming blade (SCAG, p. 142):
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.
So, with the addition of booming blade, if I have a quarterstaff + War Caster + Polearm Master, could I use my reaction to cast booming blade if someone steps into my reach?
Best Answer
RAW, it appears that using Booming Blade with Polearm Master and War Caster would work, but only with a Quarterstaff.
Polearm Master causes an enemy to provoke an Opportunity Attack; not a "melee weapon attack" which would preclude spells.
War Caster applies to Opportunity Attacks triggered by enemy movement; not "an enemy moving out of your reach", which would preclude Polearm Master's effect.
Booming Blade has a range of 5ft, targets a single enemy, and uses the weapon that allowed Polearm Master to trigger. This satisfies all requirements of both feats, as well as Mearls' answer on Twitter.
Finally, this only works with a Quarterstaff because other polearms viable with Polearm Master have a range of 10ft. When a creature enters that range and triggers the feat's Opportunity Attack, they would still be out of range of Booming Blade.
This appears to be implicitly confirmed by this Twitter exchange with Mike Mearls about whether Spell Sniper extends Booming Blade (it does, according to Mearls).