Suppose the universe is populated only with particles (or fields or strings) p1 and p2. Then p1 and p2 are the only embodied frames of reference in the universe. So suppose there is no disembodied, objective, 3rd-person notion of time that is independent of both p1 and p2. In this case, p1 evolves according to its own dimension of time t1, and p2 evolves along its own ontologically independent dimension of time, t2. There is no time parameter t that is external to p1 and p2.
Assume p1 and p2 start out as one system at t1 = 0 and t2 = 0, then they split into two particles each evolving according to their own independent dimensions of time, and interact again at t1 = 1 and t2 = 1. In the reference frame of p1, p2 does not evolve in time, i.e. t1 is not a parameter of the equations of motion of p2. Thus, p2 is not in a definite state for p1 in t1. At some time t1, say, t1 = .5, p2 is not in a classical state. If p2 were in a classical state in the frame of p1 it would be evolving according to t1, but it does not evolve in the dimension t1. And vice versa.
Then what happens with p2 from the perspective of p1? The particles interact at t1 = 0 and again at t1 = 1. But in between, p2 does not follow a classical path (as parametrized by t1). There is no information about which path p2 will have taken. Thus it could have taken any path. Before t1 = 1 all these paths are mere possibilities. That could be the origin of possibilities in quantum mechanics. And the possibilities could be origin of the additive probability amplitudes of quantum mechanics.
In the frame of p1, there is no classical path of p2 given by
because there is no such t. In fact, there is no classical functional for the combined p1 and p2 system (there is no 'time' at which both p1 and p2 have a definite state, except for the endpoints). Instead there are non-classical possibilities from each particle's frame of reference given by, one hopes, the quantum amplitudes.
To Balasubramanian's third question, "Is there a connection between the existence of a time, and the quantumness of the universe?"  the answer of this proto-interpretation is yes.
 What we don't know about time, Vijay Balasubramanian, Essay for a special issue of Foundations of Physics commemorating "Forty years of string theory" arXiv:1107.2897v1 [hep-th].