Forgive me for sounding dramatic, but the use of a leash turns out to be something of a two-edged sword. When it comes to teaching your kids to ski, it's worth taking a moment to contemplate the good and the bad of the leash so that you can use it to best effect (if you choose) while avoiding the worst of its pitfalls. Should you use a leash? As with most things involving children, the answer is: it depends.
The Argument Unleashed
In my informal survey of ski instructors who specialize in teaching young children, all were universally in agreement that leash systems should be used as little as possible (or not at all). Put the leash away, they say, as soon as the child has the leg strength and motor coordination to effectively use the wedge or 'pizza' shape. That may sound a little draconian, but there's quite a lot of logic to it.
First of all: you may never need a leash. Choose your terrain carefully, and your child may feel comfortable sliding slowly around on their own untethered from the very start. If not, holding onto your child via a harness accomplishes much of what a leash does (provided your child doesn't sag backward into your arms) without establishing the fixed connection that the leash will soon come to represent.
The whole point of skiing, after all, is to enjoy the freedom of sliding down, and to develop the ability to ski in control. A leash short-circuits this process, outsourcing the critical job of steering and stopping, eliminating the need for self-reliance. Can this lead to arrested development? Yes. In the worst cases, the child simply refuses to ski off-leash, even long after they no longer need it.
In Favor of a Connection
Life is rarely black and white. In fact, there are times when a leash can be fantastic. A leash allows you to stand upright behind your child, vastly decreasing strain on your leg and back muscles. It allows your child an introduction to balancing and whooshing on skis well before they would otherwise be able to do it on their own. As a first taste of skiing for a two-year-old, a run or two on a leash is hard to beat.
There are other benefits. Skiing on a leash gently nudges a child into a correct upper body position, with their weight centered over the skis. It requires kids to stand upright on their own, rather than leaning on you, again developing coordination. A leash allows you and your child to actually 'ski' a run together. And it establishes a physical connection that builds trust and diffuses anxiety.
Always consider both the child and the context when it comes to deciding whether or not to use a leash. Read your child's temperament—some kids cling to you, some push you away. Whatever you do, I'll echo the comments of the experts and suggest that as soon as you see the beginnings of an effective pizza shape forming, you switch to easier terrain and start working on teaching your child to start and stop on their own, untethered.
If you do go with a leash, use the simplest strap you can find, and always untie it before boarding a chairlift. Avoid complex dual-leash systems, as they are more difficult to stop using (for both of you). With any free-hanging leash, be aware that you will be skiing very close behind your child. If they fall, you'll have to stop very quickly to avoid hitting them.
Finally, remember: our goal is to get our kids skiing on their own, not on a leash. The leash can become a crutch for both parent and child because it makes things so much easier for both. If you find yourself absolutely needing a leash, that's probably a good indicator you're on terrain that is inappropriately steep for your child. Chose your terrain wisely, and you may well never need a leash in the first place, which is probably the best way to go.
Of Mountains and Molehills
First Steps: Kids on Snow
Finding Appropriate Terrain
Skiing With Kids: Dry Land Training