How to compare equality of lists of arrays with modern Java?

The for loop at least can be streamified, leading to:

return (list1.size()==list2.size() &&
        IntStream.range(0, list1.size())
                 .allMatch(i -> Arrays.equals(list1.get(i), list2.get(i)));

1) Solution based on Java 8 streams:

List<List<String>> first = list1.stream().map(Arrays::asList).collect(toList());
List<List<String>> second = list2.stream().map(Arrays::asList).collect(toList());
return first.equals(second);

2) Much simpler solution (works in Java 5+):

return Arrays.deepEquals(list1.toArray(), list2.toArray());

3) Regarding your new requirement (to check the contained String arrays length first), you could write a generic helper method that does equality check for transformed lists:

<T, U> boolean equal(List<T> list1, List<T> list2, Function<T, U> mapper) {
    List<U> first = list1.stream().map(mapper).collect(toList());
    List<U> second = list2.stream().map(mapper).collect(toList());
    return first.equals(second);
}

Then the solution could be:

return equal(list1, list2, s -> s.length)
    && equal(list1, list2, Arrays::asList);

using zip (which originates from lambda b93) function from https://stackoverflow.com/a/23529010/755183, code could look like:

boolean match = a.size() == b.size() && 
                zip(a.stream(), b.stream(), Arrays::deepEquals).
                allMatch(equal -> equal)

update

in order to check size of arrays first and then content this could be a solution to consider

final boolean match = a.size() == b.size() 
                   && zip(a.stream(), b.stream(), (as, bs) -> as.length == bs.length).
                      allMatch(equal -> equal)
                   && zip(a.stream(), b.stream(), Arrays::deepEquals).
                      allMatch(equal -> equal);

You could use a stream if the lists are random access lists (so that a call to get is fast - generally constant time) leading to:

//checks for null and size before
boolean same = IntStream.range(0, list1.size()).allMatch(i -> Arrays.equals(list1.get(i), list2.get(i)));

However, you might give as parameters some implementations that are not (such as LinkedLists). In this case, the best way is to use the iterator explicitly. Something like:

boolean compare(List<String[]> list1, List<String[]> list2) {

    //checks for null and size

    Iterator<String[]> iteList1 = list1.iterator();
    Iterator<String[]> iteList2 = list2.iterator();

    while(iteList1.hasNext()) {
        if(!Arrays.equals(iteList1.next(), iteList2.next())) {
            return false;
        }
    }
    return true;
}

You could stream over one list and compare to each element of the other by using an iterator:

Iterator<String[]> it = list1.iterator();
boolean match = list1.size() == list2.size() &&
                list2.stream().allMatch(a -> Arrays.equals(a, it.next()));

Using an iterator instead of the get(index) method on the first list is better because it doesn't matter whether the list is RandomAccess or not.

Note: this only works with a sequential stream. Using a parallel stream will lead to wrong results.

EDIT: As per the question last edit, which indicates it would be better to check the length of every pair of arrays in advance, I think it could be achieved with a slight modification to my previous code:

Iterator<String[]> itLength = list1.iterator();
Iterator<String[]> itContents = list1.iterator();

boolean match = 
        list1.size() == list2.size()
    && 
        list2.stream()
            .allMatch(a -> {
                String[] s = itLength.next();
                return s == null ? a == null :
                       a == null ? s == null :
                       a.length == s.length;
            })
    && 
        list2.stream()
            .allMatch(a -> Arrays.equals(a, itContents.next()));

Here I'm using two iterators and am streaming list2 twice, but I see no other way to check all lengths before checking the contents of the first pair of arrays. Check for lengths is null-safe, while check for contents is delegated to the Arrays.equals(array1, array2) method.