I'm pretty sure you're going to need a few more libraries than just qtcore. First, figure out what you ACTUALLY need:
This should dump a list of libraries. ALL of the listed libraries have to be available on the target system. (If these libraries dynamically load other libraries with dlopen(), you will not know what those libraries are until it fails at runtime)
If you copy all of these libraries into the same directory as the program itself, it MIGHT work, if you start the program this way:
Code:
$ LD_LIBRARY_PATH=.:$LD_LIBRARY_PATH ./MyProg