What prevents Bitcoins for being used to avoid taxation? Today, states can follow money trails and transaction histories, usually involving checking bank accounts. As as I understand, It's impossible for anyone else to read your Bitcoin transactions and current balance without your consent.

In a Bitcoin world, how can a state monitor bank account when it's impossible to check their transactions?
What prevents Bitcoins for being used by criminals to avoid
  taxation?


nothing.  thats the price of anonymity:  it can be used for good and for bad.  


  In a Bitcoin world, how can a state tax its citizens when it's
  impossible to check their transactions?


in a hypothetical purely bitcoin world, the state would have to find different ways of funding itself, if it existed at all.