Some open source tools one can use for test management: -
TestLink is an open source test management tool which will help you synchronise requirement and test specification together. With this, you can create test project and also document test cases. If you have multiple people working on different user roles this can prove to be of help.
2. Squash TM (Test Management)
Squash TM is open source test repository manager which is found in Squash toolkit. With this, you can manage requirements, campaigns, and test cases execution.
3. Kiwi TCMS
It is the leading open source test case management system (TCMS) which can help you manage your test cases, plans and runs efficiently and also helps you improve testing productivity & easy reporting. It also provides integration with popular issue tracker easing your process.
Though this is not an open source tool but it is one of the best alternatives to look for when you talk about test case management tools. You can try it on cloud as well as download it in your local machine. It helps you create a project, you can add test cases with their steps, start them, add results that you get, integrate with other tools like JIRA, Redmine and invite your team members.