How can I compare two time strings in the format HH:MM:SS?

As Felix Kling said in the comments, provided your times are based on a 24 hour clock (and they should be if there's no AM/PM) and provided they are always in the format HH:MM:SS you can do a direct string comparison:

var str1 = "10:20:45",
    str2 = "05:10:10";

if (str1 > str2)
    alert("Time 1 is later than time 2");
else
    alert("Time 2 is later than time 1");

Date.parse('01/01/2011 10:20:45') > Date.parse('01/01/2011 5:10:10')
> true

The 1st January is an arbitrary date, doesn't mean anything.

Date object in js support comparison, set them same date for compare hh:mm:ss :

new Date ('1/1/1999 ' + '10:20:45') > new Date ('1/1/1999 ' + '5:10:10') 
> true