I don't believe a protocol can enforce weak-ness. For example:

protocol ObjectProtocol: class {
  weak var weakProperty: AnyObject? {get set}
}

class ObjectImpl1: ObjectProtocol {
  weak var weakProperty: AnyObject?
}

class ObjectImpl2: ObjectProtocol {
  var weakProperty: AnyObject?
}

These both compile ok, even though the protocol has weak but ObjectImpl2 does not implement it.

EDIT: Is this what you're after?...

protocol ObjectProtocol: class {
  typealias WeakPropertyType: Any //must be a class type
  var weakProperty: WeakPropertyType? {get set}
}

protocol WeakPropertyProtocol: class {}

class ObjectImpl: ObjectProtocol {
  typealias WeakPropertyType = WeakPropertyProtocol
  weak var weakProperty: WeakPropertyProtocol?
}

This implementation requires use of Any rather than AnyObject, since WeakPropertyProtocol is a protocol rather than a class.

Or this?...

protocol WeakPropertyProtocol: class {}

protocol ObjectProtocol: class {
  typealias WeakPropertyType: AnyObject //must be a class type
  var weakProperty: WeakPropertyType? {get set}
}

class MyWeakClass: WeakPropertyProtocol {

}

class ObjectImpl: ObjectProtocol {
  typealias WeakPropertyType = MyWeakClass
  weak var weakProperty: MyWeakClass?
}

Either way, I think the key is in defining which class/protocol to use for WeakPropertyType.