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Would you add your proficiency bonus to a pebble from the magic stone cantrip? The spell description says nothing about it.

The magic stone spell description states (EEPC, p. 20; XGtE, p. 160):

You touch one to three pebbles and imbue them with magic. You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling. If thrown, it has a range of 60 feet. If someone else attacks with the pebble, that attacker adds your spellcasting ability modifier, not the attacker’s, to the attack roll. On a hit, the target takes bludgeoning damage equal to 1d6 + your spellcasting ability modifier. Hit or miss, the spell then ends on the stone. If you cast this spell again, the spell ends early on any pebbles still affected by it.

Would this be considered an improvised weapon that wouldn't get the proficiency bonus added to the attack?

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Yes, you're making a ranged spell attack

Magic stone (Elemental Evil Player's Companion, 160) states:

You [...] can make a ranged spell attack [...]

All creatures are proficient in [ranged] spell attacks, so you are proficient in this one as well. Chapter 10 of the PHB covers this:

Your attack bonus with a spell attack equals your spellcasting ability modifier + your proficiency bonus.

However, magic stone has the additional caveat that you use the original caster's modifier:

If someone else attacks with the pebble, that attacker adds your spellcasting ability modifier, not the attacker’s, to the attack roll.

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Thrower's Proficiency + Caster's Ability

The character throwing the stone (either by hand or using a sling) adds their proficiency bonus and the caster's ability score to the attack roll.

…that attacker adds your spellcasting ability modifier, not the attacker’s, to the attack roll.

The spell specifies "ability modifier"; not "ability modifer and proficiency bonus."

For example:

Casey the Wizard (who cast magic stone) has an INT of +5 (20) and a proficiency bonus of +3.

Tamryn the Thrower has a DEX of +2 (14) and a proficiency bonus of +4. They are not proficient in a sling.

Tamryn throws a stone. Ranged Spell Attack: 1d20 +4 (Tamryn) +5 (Casey) to hit. Range 60ft. Hit: 1d6 +5 (Casey) bludgeoning.

Tamryn uses a sling. Ranged Spell Attack: 1d20 +5 (Casey) to hit. Range 30/120ft. Hit: 1d6 +5 (Casey) bludgeoning.

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