The efficiency depends on the query size relative to the data distribution. A small query in a sparse region prunes almost everything. A query that covers the whole space prunes nothing (because every node overlaps), degenerating to a brute-force scan. The quadtree gives you the most benefit when your queries are spatially local, which is exactly the common case for map applications, game physics, and spatial databases.
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The library project, on the other hand, contains the actual core gameplay logic. In the inventory service's case, this includes the C# classes that model the request and response for every API operation as well as the underlying logic for handling the countless inventory related systems such as crafting, quests, loot, etc. As a separate project, the library can be imported in other services when we need to reuse that logic without pulling in any of the web project's code.