How can the universe evolve unitarily if there's no clock outside it? We say that the wavefunction of the universe always evolves unitarily as everything gets entangled. But there is a huge problem with this reasoning. A clock has to exist outside the universe for us to define unitary evolution of the universe. But there's no clock outside the universe.
It is problematic that the formulations of quantum mechanics rely on the existence of "periodic phenonemena" like clocks, even when such phenomena do not really exist at the fundamental level. If light had been a periodic wave, then we could always refer to it to define our clocks. But light itself is quantum. Since everything is quantum, we can't have clocks at the fundamental level. Clocks only exist as approximate macroscopic phenomena.
So once again we're running into the problem of us defining the microscopic world using macroscopic phenomena, which themselves are an approximation of the microscopic world. This is just like how the definition of collapse also relies on macroscopic phenomena.
So, how can we define quantum mechanics without relying on macroscopic notions like clocks?
 A: 
We say that the wavefunction of the universe always evolves unitarily as everything gets entangled. But there is a huge problem with this reasoning. A clock has to exist outside the universe for us to define unitary evolution of the universe. But there's no clock outside the universe.

There is no clock outside the universe by which to judge whether evolution is unitary. The solution to this problem is that the universe as a whole is in a stationary state of a suitable Hamiltonian. In that state there is a clock observable and the relative state of the rest of the universe evolves unitarily with respect to the values of that clock observable. For more details, see
Evolution without evolution: Dynamics described by stationary observables
Evolution without evolution, and without ambiguities
A: You are assuming that time does not exist without clocks. That is analogous to assuming that space does not exist without rulers, and both assumptions are unjustified.
As far as we know, we live in a four dimensional spacetime which exists whether or not there are any clocks or rulers to quantify its dimensions.
A: I'm not sure an answer exists to your question because it is not stated sufficiently precisely. However I think it is worth pointing out that we need to distinguish between the time coordinate and the flow of time. The former is a very simple concept in physics while the latter does not exist at a fundamental level.
You use the example of a light wave, and we can write such a wave as:
$$ E(t,\mathbf x) = E_0 \sin(\omega t + \mathbf k\cdot \mathbf x) $$
Here the function describing the light wave is a function of two variables, $t$ and $x$, and it is periodic in both so why would we say time flows but space doesn't? Both are just coordinates we use to locate points in a four dimensional manifold with three spatial axes and one time axis. There is no concept of flow implicit in our equation.
When you talk about a clock this is just a correlation between two systems. For example we could have some complex system, $F(t,\mathbf x)$ that we are trying to time. We could use our light wave as a clock, but this just means we are comparing how the variation of $F$ with $t$ compares to the evolution of $E$ with $t$. We could equally compare the evolutions of $F$ and $E$ with $\mathbf x$. This is mathematically just as valid but implies no flow of the space.
Obviously we all experience time as flowing, at one second per second, and this makes it fundamentally different from space, but it is not clear that this is anything more than some quirk of how human consciousness works. This concept of flow does not exist at a fundamental level.
So I think your question is not well founded as we do not need external clocks in the universe. There is just a time axis and for any system we can integrate along this axis to calculate how a system depends on the $t$ coordinate just as we can integrate along the spatial axes.
A: It's a rather odd assumption why the universal evolution should care about a clock outside of it in the first place. Time can be defined as the unidirectional progress of irreversible processes. This evolution, be it an ever increasing of number of superimposed states evolving, an interaction-induced collapsing wavefunctions (the mechanism of which is unknown), increasing entropy, or an expansion of the universe happens regardless of imaginary ideal clocks or real approximate clocks measuring it. You can put them besides the processes constituting time or you can do that in thought, but the process itself won't care about that.
Maybe the only true periodic process is found in the vacuum itself. The omnipresent virtual particle field can be described mathematically as a superposition of all independent energy and momentum states going back and forth in time simultaneously. So in a sense they constitute ideal periodicity. From this (virtual) state real unidirectional irreversible particle processes can emerge.
