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The spell Melf's Acid Arrow does 4d4 on hit and 2d4 next turn (I'm ignoring what happens for a miss). If you roll a 20 for your spell attack roll, you double the 4d4 to 8d4.

What about the 2d4 next turn? Does that get doubled too or does that just stay at 2d4 because it occurs on the next turn?

I know that How does extra damage work for critical hits? covers additional kinds of damage during a critical hit, but this is the same substance that did critical damage, continuing its damage in a later round.

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Only the part that is contingent on the attack roll itself is doubled

There seems to be intent that it is not doubled, as evidenced by the Green Flame Blade question below:

In a tweet Crawford specifically answers a question about the subsequent damage on Green Flame Blade.

Question:

If you critically hit with the attack part of Green Flame Blade, and are greater than level 5, do you roll extra dmg against the 2nd target?

Answer from Crawford:

The splash damage of green-flame blade isn't affected by the attack critting. Think of the attack as process X & the splash damage as Y

Coupled with Sage Advice 2020 p13:

Can spell attacks score critical hits? A spell attack can definitely score a critical hit. The rule on critical hits applies to attack rolls of any sort.

I do realize that JC's Tweets are no longer dogma but until definitive clarifications come from Sage Advice they will have to suffice. That said JC's Tweet on Witchbolt criticals dovetail nicely with Melf's Acid Arrow in so far as the adjudication of damage on subsequent turns.

Question:

What if the Witch bolt crited? Since there's no attack roll for the subsequent damage, would it roll only the first as a crit or all of them as crits? (I assume only the first one, but eh)

Answer:

Witch bolt's initial attack can, indeed, crit, but the damage delivered via actions on subsequent turns is unaffected by the crit. A crit isn't intended to affect the damage of other actions/reactions/attacks.

These lead me to believe that the only part of a spell attack that is doubled is the part that is dealt as part of the actual attack roll. As the subsequent round's damage would be the 'Y' or even 'Z' in this instance.

I went through the podcasts and couldn't find anything else, mostly the spells section dealt with targeting in the one from Jan 17, 2017.

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It is not doubled.

The rules for critical hits state:

When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together.

Melf’s acid arrow says:

Make a ranged spell attack against the target. On a hit, the target takes 4d4 acid damage immediately and 2d4 acid damage at the end of its next turn.

The damage for the hit is 4d4 acid damage. The damage at the end of the next turn is damage conditioned on the hit, not the attack’s damage. The extra damage is not the attack’s damage, as it happens next turn, not as a part of resolving the attack. The attack’s damage is those damage dice that are rolled as part of resolving the attack:

Resolving the attack. You make the attack roll. On a hit, you roll damage.

The extra damage is not part of resolving the attack, so it is not “the attack’s damage”. Since the rules for critical hits state that only damage dice for the attack are doubled, the extra damage is not doubled.

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It gets doubled

The rule for critical hits is:

When you score a critical hit, you get to roll extra dice for the attack’s damage against the target. Roll all of the attack’s damage dice twice and add them together.

The 2d4 are part of “the attack’s damage dice” so they get rolled twice.

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