Can I view a memory before the Captured Memories Quest?
I stumbled upon the Lanayru Road East Gate memory location prior to completing the Locked Mementos quest. I have collected the camera rune from Purah, now the quest states:
You proved the Camera works again by capturing an image of Purah! According to Purah, the album already has pictures from the past in it. Where could these have come from? Impa in Kakariko Village may have the answer.
I was wondering if I would be able to view the memory, however, no memory shows up. See image below:
I then spoke to Impa to complete the quest and receive the Captured Memories quest and sure enough, the memory is available. 
Do I have to complete the Locked Mementos quest prior to being able to view the memories? If this is the case, I just think it is unfortunate that you have to back track to Kakariko village before being able to see the memories even though the memories are on the Sheikah Slate. If this is not the case, are the memories available at certain times?
Best Answer
Sometimes
The "yellow sparkles" memories, like the one pictured, are not accessible before obtaining the Captured Memories quest.
However...
During all four of the Divine Beast quests, you recover a memory of the corresponding Champion. These do not count towards Captured Memories and therefore also do not require Captured Memories to be started. Also, in the Champions' Ballad DLC, after hearing each Champion's song, you will recover another memory of the Champion. Finally, at the end of the Champions' Ballad DLC, after Kass plays the Champions' Ballad, you will recover one last "non-Captured Memories" memory (EX Recovered Memory #5 - The Champions' Ballad).
Pictures about "Can I view a memory before the Captured Memories Quest?"
Is there a way to rewatch Memories BotW?
You have to go to your Adventure Log (the screen left of your inventory) and scroll over to the Memories page. There you'll be able to revisit any restored memories.Do you have to do the captured Memories quest?
The Captured Memories quest can span the entirety of The Legend of Zelda: Breath of the Wild. There's no timer, and it's not a requirement to complete the game. But those who complete the quest will reap its rewards.Can you get Memories before seeing Impa?
Unfortunately, yes. The idea is that Impa is the one helping you to restore your memories, not Purah.How do you view Memories in Zelda?
In order to find and view BotW Memories, you must search for the locations based on the visual clues in your Sheikah Slate Album. Pikango the painter, found at various stables and towns throughut Hyrule, can offer additional clues. He is found at the town or stable closest to the memory on the map.Sources: Stack Exchange - This article follows the attribution requirements of Stack Exchange and is licensed under CC BY-SA 3.0.
Images: Harrison Haines, Anete Lusina, Anete Lusina, Amina Filkins