Passive - Each time X's spell hits an enemy, it applies a stack onto them. Once X manages to get 3 stacks on the enemy units, her next spell with have an increased affect and halves the cooldown of the ability she uses on them. She cannot stack for 8/7/6 seconds after she has activated the stacks. If she does not use a spell on an enemy unit with stacks for 4/4.5/5 seconds, the stacks will disappear one by one each .5/.75/1 seconds.
Interesting take on the "three-hit-bonus" concept. There are a lot of them in League, though. xD However, I'd change the stacks to 2, with the third immediately adding the bonus effect. Needing 4 spells means that realistically, you're only going to be able to use your passive on Q (because W and E's Cooldowns are way too high - you'll cast W, 1 stack, E, 2 stacks, Q, another stack... and then you have to wait up to 6 seconds for your Q to come back, which means you're not gonna get any bonus effects whatsoever - as such, the only way to get the 4th stack is to start with Q).
That is, unless you increase the duration of the stacks to, like, 10 seconds or something.
Q - X sends out a shard of ice dealing damage and snaring the first enemy hit for 1/1.25/1.5/1.75/2 second.
Q with Passive - Shard can pass through stacked enemy unit and hit an additional enemy unit. Stacked targets are stunned instead of snared(this also applies onto the additional unit).
Magic Damage: 60/85/100/125/150 (+40% AP)
Cooldown: 12/11/10/9/8 8/7.5/7/6.5/6
Mana Cost: 60/65/70/75/80
Generic skillshot snare spell. I mean it's a good idea on paper, but it's so similar to other spells already that are basically "shoot projectile, CC first/multiple target(s) hit".
W - X blows cold wind towards an area, slowing enemies inside it for 8 seconds. Enemy units inside the area are slowed for 50%/55%/60%/65%/70%. Enemies who enter/leave the area get 1 stack(CAN ONLY APPLY ONCE). If enemy units stay in the cold area for 8/7/6 4 seconds, they are stunned for 1.5 second.
W with Passive - (Players that enter the area with 3 stacks: Slow gets stronger every .5 seconds. Maximum of 2 seconds.) (Players that leave the area with 3 stacks: Remain slowed for 2 seconds.)
Cooldown: 20
Mana Cost: 80/90/100/110/120
Bro this is basically Viktor's W. xD Except his slow is [28/32/36/40/44]%, the zone lasts only 4 seconds, and it takes 1.5 seconds for an enemy standing inside to be stunned for 1.5 seconds. It fits this champion's kit, though.
E - X forms a shield around an ally champion for 8 seconds. The ally champion is shielded for 80/120/160/200/240 (+60% AP) Any enemy unit attacking the shielded ally are slowed for 30% (+15 AP). If an enemy champion destroys the shield, they are stunned for 1/1.25/1.5/1.75/2 seconds.
E with Passive: If the shield is destroyed BY AN ENEMY UNIT, the ally champion gains a burst of movement speed.
Movement Spd: 50%/55%/60%/65%/70%
Cooldown: 12 10
Mana Cost: 60/70/80/90/100
I don't like this spell mainly because it punishes you for doing something well. Honestly, this much utility on a simple shield is Ultimate-worthy.
R - X summons a huge blizzard in an area dealing 50/70/90 (+20% AP) for 10 seconds. Enemies inside in the area are slowed for 20%/30%/40% while allies are speed up for 10%/20%/30%.
Cooldown: 120/110/100
Mana Cost: 100/110/120
I actually kind of like this spell, if it didn't overlap with Anivia's (and to a lesser extent, Gangplank's). While they're intrinsically different (one lasts forever as long as you have the mana, the other has a fixed duration), both of them have the same goal: a relatively low damage slowing zone that your opponent should try to avoid. And honestly, Anivia's is just better. I'd rather have a smaller, low Cooldown zone control spell than a larger high cooldown one, even if it's, like, huge (and a giant-sized slow zone that can buff allies is Gangplank's thing, anyway).
Because of how they work, I tried flipping your E and R around.
E - X summons a blizzard in an area, dealing [110/170/230/290/350] (+100% AP) magic damage over 4 seconds, with each tick of damage occurring at 0.5 sec. intervals. Enemies inside in the area are slowed for [20/25/30/35/40]% while allies are speed up by [10/15/20/25/30]%. Enemies that take damage from this spell only get 1 stack of the passive.
E with Passive: Enemies that enter the field with 3 stacks have their MR shredded by 4% for every tick of damage (up to a maximum of 20% MR reduced). Those that leave with 3 stacks are unaffected.
Cooldown: 12 seconds
Mana Cost: 90
With 40% CDR, you can almost double-cast E if you hit 3 stacks, since the Cooldown drops to 3.6 seconds, meaning you can use it again before it even ends (though it'll cost you a lot of Mana). Now it's more unique as a zone-control spell, and the 3 stack effect has added utility in MR shredding. Plus, you now you can actually combo your spells together.
R - X starts channeling ethereal frost magic for 1 second, and then summons an enchanted barrier made of ice on target ally champion, shielding them for 300/500/700(+90% AP). Any enemy unit attacking the shielded ally have their movement speed slowed by 30% (+5% per 100 AP) and their attack speed slowed by 15% (+5% per 100 AP). If shield is not destroyed before it expires, nearby enemies are dealt [100/175/250] (+60% AP) Magic Damage and are stunned for 2 seconds. Can be cast on self.
Cooldown: 100/90/80 sec
Mana Cost: 100/100/100
I changed it so that it stuns ALL nearby enemies if you DON'T break the shield. Honestly, it makes no sense that you're punished for breaking a shield. That's just really counterintuitive, and, for lack of a better term, toxic gameplay. However, I kept the Randuin's-like effect of slowing those that damage the shield. That part is pretty cool, so it should stay on. Also, it's not actual channeling (like Fiddlesticks), so you can't cancel the spell, but because of how strong the spell is, I added a windup time (think like Ezreal's ult).
I tacked on some damage, too. Since your W doesn't do any damage, I figured, why not? Plus, the damage isn't too high (it's much lower than Lissandra's ult, for example). However, the damage can be removed if necessary (I'm assuming your champion's a support, right? So too much damage can be an issue)
Well, that's my 2 cents on this. You don't have to agree - I'm just throwing my ideas out there.