Answer:
John used the ginger.
Explanation:
Consider the provided information.
When the pie is finished, there is 1/4 cinnamon left, 1/3 of nutmeg, 1/2 of ginger and 5/8 of pumpkin left.
First arrange the them in increasing order.

It is given that Quincy used the most ingredient, that means Quincy left with least amount of ingredients. The least number among them is 1/4 which is cinnamon.
So Quincy used cinnamon.
It is given that Lucy used the least ingredient, that means most of the her ingredients were left which is pumpkin i.e 5/8
So Lucy used pumpkin.
It is given that Prudence used more ingredient than John, that means Prudence left with the less ingredients as compare to John which is nutmeg i.e 1/3
Thus, Prudence used nutmeg.
Hence, John used the ginger.