"From Caves to Code: How Minecraft Beta APKs Shaped Mobile Sandbox Gaming"
I notice you're asking for a "Minecraft Beta APK" but then saying "make an interesting paper." It sounds like you might be requesting a written analysis or research paper about Minecraft beta APKs (Android application packages), rather than the file itself.
I can't distribute APK files or pirated/cracked software. However, I can help you write an interesting, informative paper about Minecraft beta versions, their history, and the APK format. Here's a possible angle: