mersenne twister - is there a way to jump to a particular state?

Yes it is possible! It's called Jump Ahead.

You can find all the details to do this with Mersenne Twister on the homepage of MT's authors. Code is available as well as scientific publications explaining the algorithm:

http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/JUMP/index.html

Today, there is a way to do this using discard. The complexity is linear in number of equivalent advances.

A working example:

  std::mt19937 engine(0);
  std::uniform_int_distribution<int> dis(0, 9);
  auto generator = std::bind(std::ref(dis), std::ref(engine));

  qDebug() << "First sequence: ";
  for (int i = 0; i < 20; i++) {
    qDebug() << "Random value: " << generator();
  }

  engine.seed(0); // reset
  engine.discard(15); // discard the first 15 numbers from the first sequence
  qDebug() << "Last five numbers from first sequence: ";
  for (int i = 0; i < 5; i++) {
    qDebug() << "Random value: " << generator();
  }

The Mersenne Twister can be represented as a (rather large) matrix over F₂ (the field containing the two elements 0 and 1). A transition to the next state is a multiplication by this matrix.

To jump to some arbitrary position in the stream you can compute the corresponding power of this matrix by repeated squaring, and multiply it with your initial state.