If Fading Memories gives a unit temporary and it dies at the start of its controller's beginning phase, does it trigger a Hidden Sprite call? If so, does the called sprite also die due to its own temporary, or does this trigger the next round?

Yes, you can use a hidden Sprite Call to react to a unit dying from Temporary. Because Temporary creates a triggered ability that goes on the chain at the start of your turn, you have the opportunity to respond to it.

Here is how the interaction plays out, based on the rules and common game sequences (referencing RiftJudge FAQ #10474 and #9228):

The Sequence

1. Beginning Phase: The Temporary ability of your unit (granted by Fading Memories) triggers and is placed on the chain as a pending item.
2. Reaction: Because an item is on the chain, you may play your hidden Sprite Call as a reaction. It is placed on top of the chain.
3. Resolution (LIFO - Last-In, First-Out):
   • Sprite Call resolves first: You summon a new Sprite token with Temporary to the battlefield.
   • Original Temporary trigger resolves: The original unit dies.
4. Result: The new Sprite token entered the battlefield after the Temporary trigger for that specific turn had already been processed.

Does the new Sprite die?

No, the new Sprite will not die this turn. Because it entered the battlefield after the game already checked for Temporary triggers during that turn's Beginning Phase, it is "safe" until the start of your next turn.

Important Clarifications

• Hidden Restriction: Per Rule 811.1.d.3, because you are playing Sprite Call from a hidden state, the new Sprite token must be summoned to the same battlefield where Sprite Call was hidden.
• LIFO Resolution: This interaction relies on Rule 336.1 (Last-In, First-Out). The reaction resolves before the original trigger, allowing the new unit to avoid the "death check" that just occurred.