Alright, let's get Firebase Firestore up and running. It's a cloud-hosted NoSQL database, perfect for storing and syncing data. First things first, head over to the Firebase Console and create a new project. Once that's done, you'll need to add the Firebase Core and Firestore dependencies to your Maven project.

For your web app, just pop this script in:

<script src="https://www.gstatic.com/firebasejs/7.24.0/firebase-app.js"></script>
<script src="https://www.gstatic.com/firebasejs/7.24.0/firebase-firestore.js"></script>

Next, you'll need to initialize Firebase in your app. Use the configuration code you got from the Firebase Console for this step.

Now, let's create a Firestore database. In the Firebase Console, start a new Firestore database, pick either test mode or production mode, and choose a location. Then, create a collection to store the leaderboard data. This collection will have documents that uniquely identify each player's name and score.

Here's how you can create documents in your Firestore collection:

var db = firebase.firestore();
var newScore = db.collection("scores").doc();

When a player's score updates in your app, you can instantly write this data into Firestore using the .set() method. For example:

newScore.set({
    name: "player1",
    score: 100
})
.then(function(docRef) {
    console.log("Document written with ID: ", docRef.id);
})
.catch(function(error) {
    console.error("Error adding document: ", error);
});

So, every time a score updates, it will be reflected in your Firestore database.

To show the leaderboards, you need to fetch the score data from Firestore using the .get() method. Then, you can display this data in your app.

db.collection("scores").orderBy("score", "desc").get().then(function(querySnapshot) {
    querySnapshot.forEach(function(doc) {
        console.log(doc.id, " => ", doc.data());
        // Display the data in your application
    });
});

This will fetch all the data from the "scores" collection and order it in descending order based on the "score" field.

Firestore also supports real-time syncing. This means whenever a new document is added to the "scores" collection, Firestore sends an event to your app with the updated data. You can keep your leaderboard updated in real time by listening to these events. Here's how:

db.collection("scores").orderBy("score", "desc")
.onSnapshot(function(querySnapshot) {
    var scores = [];
    querySnapshot.forEach(function(doc) {
        scores.push(doc.data());
    });
    // Update the leaderboard
});

This will keep your leaderboard data synced in real time, updating every time there's a change in the 'scores' collection. All new, modified, or removed scores will immediately reflect in your leaderboard.