What's stopping someone from taking over all the bitcoin addresses? Anyone can generate unlimited bitcoin addresses.

Presumably, there's a limited number of these (even if it's a very very large number).

What's stopping someone from taking over all the bitcoin addresses? (If the answer is current computer speeds, is it something that's possible in a future 'quantum computer' type era?)
Presumably, there's a limited number of these (even if it's a very very large number).


Yes, this is true.


  What's stopping someone from taking over all the bitcoin addresses?


Nothing. You can even guess someones private key and steal all their savings.

What you underestimate is how big a number can be. Every digit you add to your number multiplies number of tries you have to perform by 10. Computers might seem fast but they live in same realm, same rules apply to them. Very quickly you run into such big numbers that even biggest computing efforts on this planet can't keep up. Then you go further and when you have number you feel secure with, you stop.

This might seem stupid, because someone might get lucky and gain access to your "number". Yet we don't think twice about very simple passwords we use every day, for all our savings. Even if you have VERY complex password, it is likely nowhere near how big that number is. So we should perhaps start from improving our password practices a bit... 

Simply put, this number is so unlikely to be guessed that it's too small for us to understand, much like understanding how many transistors processors have.


  is it something that's possible in a future 'quantum computer' type era?


Quantum computer isn't actually faster than normal computer. It's just that it can solve some problems faster. And when we get quantum computers, bitcoin will be broken because it doesn't use cryptography that can stand quantum computing. Those numbers (or rather structure they compose) will be easily reversible from publicly known information. So quantum computer will break bitcoin, but we don't even know if true quantum computer is possible (and no, we don't have quantum computer, it's just a easily-catched topic by media when somebody says so).