  In this paper we define new Monte Carlo type classical and quantum hitting
times, and we prove several relationships among these and the already existing
Las Vegas type definitions. In particular, we show that for some marked state
the two types of hitting time are of the same order in both the classical and
the quantum case.
  Further, we prove that for any reversible ergodic Markov chain $P$, the
quantum hitting time of the quantum analogue of $P$ has the same order as the
square root of the classical hitting time of $P$. We also investigate the
(im)possibility of achieving a gap greater than quadratic using an alternative
quantum walk.
  Finally, we present new quantum algorithms for the detection and finding
problems. The complexities of both algorithms are related to the new,
potentially smaller, quantum hitting times. The detection algorithm is based on
phase estimation and is particularly simple. The finding algorithm combines a
similar phase estimation based procedure with ideas of Tulsi from his recent
theorem for the 2D grid. Extending his result, we show that for any
state-transitive Markov chain with unique marked state, the quantum hitting
time is of the same order for both the detection and finding problems.
