I'll base my answer on reducing the entropy of a system by reducing its temperature. So instead of trying to achieve 0 entropy, I'll try to achieve 0K temperature, which is the same thing.
One can do this cooling process only using temperatures that are 0K or greater. If you do this with objects that have 0K, well you will already have something that has 0K so the job is done! If you try with temperatures greater than 0K you will have to know the following.
The most basic process of cooling something that has temperature $T$ is to put it in contact with something that has temperature $t<T$. Now, $t>0$ since negative temperatures are not the case of this discussion, and $t=0$ was considered above. Cooling by using any $t>0$ and $t<T$ will give a final temperature $t<\hat t<T$. In order for this to work, you will have to apply this method of cooling an infinite number of times before you get to 0K. So the equivalent in entropy is that you need an infinite number of processes to get to 0 entropy.
Now, for the perfect crystal statement. A perfect crystal is one in which the atoms do not move at all. Nonmoving atoms mean a temperature that is 0K. So a perfect crystal is something that has its temperature 0K.
So you have a definition by comparison for 0 entropy, that is a perfect crystal at 0K, and the statement that you cannot reach that state unless you repeat your process an infinite number of times. Based on this the equivalence is up to you.