This is a short scene, but it’s a huge scene when it comes to important information. We get a glimpse of the time-turner that causes all of the timeline disasters in this play. We finally get a mention of Hugo! We get the official titles of Harry and Hermione’s position in the Ministry. And we get a good bit of foreshadowing hidden in Harry’s paperwork.
It would be really easy to make this minute about Hermione being Minister for Magic. But, I feel that I can’t really add anything to any existing discussion about this, so all I’ll say is that it really took me by surprise, I didn’t think it fit her, at first. Rereading the main series again made me change my mind, specifically how passionate Hermione got with S.P.E.W., this is what made me think that she could one day become Minister for Magic.
What I’d really like to discuss in this minute is the information found in Harry’s paperwork. A casual viewer of this play, (and by that I mean someone who came to watch it because they are a fan of Broadway plays, not necessarily because they are a Harry Potter fan), they probably don’t know that the things Hermione mentions reading in the mess on Harry’s desk are classic ingredients to Harry Potter stories. These hints are what Potter fans look for to help figure out the mystery within the story.
Upon this reread I couldn’t help but see the similarities between the information about the trolls, graphorns, giants and werewolves and the information that Dumbledore puts forward as evidence that Voldemort had returned (the disappearance/deaths of Bertha Jorkins, Frank Bryce, and Barty Crouch). Also, all of the memories that he compiled to learn more about Tom Riddle and his Horcruxes. The act of looking for clues in the mundane details that may seem harmless on their own, but when strung together reveal a larger picture is what Dumbledore did best and tried to pass on to Harry. 
On that note, I wanted to discuss the fact that Harry not being the one to pay attention to these details, but had to hear them from Hermione is the most accurate scene so far as far as canon of the original series goes. Harry was not an outstanding student, by any means. Hermione was usually the one to reveal information needed.
On to the positive bit from this scene: I like that Harry uses this way to get in and out of the Ministry. The visitor entrance. Here’s my second piece of Cursed Child headcanon: I’m choosing to believe this was an intended choice of Harry’s once he became a Ministry official, rather than it being another breach of canon. This was the way he first entered the ministry, and this is the way he prefers to come and go now as an adult.