I sympathize with you. But why does separating them only work for a short time? Separate them again - without yelling (since that's what they're doing). And again. And again. The 3 year old only needs 3 minutes to get the message (that's a common guideline for kids, 1 minute per year of age). The 3 yo should be reintroduced to her sister after 3 minutes, and if the bickering starts again, separate them again. The 6 yo is fine with it, probably because she gets to do something she likes and do it in peace. So that means to not let her be someone she's enjoying. Don't leave her in front of the TV or send her to her room if there's something fun to do in there (like a TV or a computer or a snack).
The immediate goal is not to get them to play nicely. The goal is to get the back behavior and the noise and the whining to stop. So what they get sent to do has to be a lot less fun than what's at hand. This will drive you absolutely nuts for about 3 days. It will be harder for you than for them. But don't give up - be insanely consistent. You won't get anything else done and that's okay.
They are getting a payoff for this behavior - they are getting attention from the other one, and from you. It's possible that they are getting to do something fun by themselves - sometimes parents get them each something (toy, activity) that will shut off the noise - which teaches them that they get something good if they fight. That's what you have to take away.
You separate them, you take any anything good from their rooms so that the rooms are spartan. My advice is to let them keep books and anything really special that comforts them (stuffed animal, special blanket, etc.). Definitely take away anything breakable so nothing suffers from their initial frustration for the first few days.
Take away anything enjoyable - even if it means packing things up and putting them in the attic! Even if the fight is about who had something first, it doesn't matter - if they are fighting and whining and touching each other, they both lose that object or TV show or whatever it is. Make SURE that you tell them THEY caused this - it's not Mean Mommy, it's their choice to not handle things fairly, gently, reasonably quietly. THEY can't get along, THEY can't follow rules. But Mom is following the rules because the rule is, if you fight, you go to your room.
Try to understand where they are coming from. The 6 yo just came from a classroom of 20 kids in which she had to share, stay in line, and be just one of the group. She comes home to being the big kid, the one who has not been there all day, and with a 3 yo waiting for her desperately. She has to adjust to a 3 yo way of thinking. She just wants to take what she wants and play with it. Meantime, the 3 yo has had your attention all day, and has her big sis come in and now suddenly there's someone new to hang out with but who wants to have certain toys. Suddenly the 3 yo has to share, so she's going to take what she wants and hold on to it. So you have immediate conflict.
Your goal has to be to stop the yelling every time one of them doesn't get her way. Which means you have to suck it up and not yell about this - which is going to be hard! Just be businesslike, giving them as little attention as possible while you achieve your goal (separation). Leave them in separate, no-fun rooms for 3 minutes. Bring them back out, but remove the one item they were fighting over, and leave it in your possession for a day. Tell them "no yelling" or "no whining" (be very specific, nothing like "you need to get along" because it's too vague for the 3 yo especially). If it starts up again, say a very concise but stern comment like "no touching - hands off!" and separate them again. 3 minutes later, out they come. Do not engage either of them for 3 minutes. And so on. After about 4 times, they will either stop, or one of them will melt down completely. If they stop, great. If there's a meltdown, then they stay in the room a lot longer. Use a kitchen timer if you have to, put up high where they can't reach it.
Now, once calm is achieved, it may be that the 6 yo needs a little re-entry time, so if that's the case, just schedule a calm snack at the table when she first comes home, with the 3 yo there, but then each of them gets to tell the other something about their day so far. Then they get to find an activity they can enjoy together, such as the 6 yo showing her work or the 3 yo saving a project or picture to give to her sister at that special time. Or they can play separately, which is fine and at least peaceful.