Can quantum randomness be somehow explained by classical uncertainty? In quantum mechanics, the outcome of each measurement is random, distributed according to the squared amplitude of the wave function obtained from the Schrodinger's equation. Now, can someone suggest that QM measurement outcomes are just produced by a deterministic, real, local process (or field) in the 3-D space, varying because of uncontrollable phenomena like noise? 
(An example of such a  classical field can be a real 3-D coherently rotating vector field accounting for the Schrödinger equation and electron spin in a consistent manner, which carries distributed angular momentum and energy in the same way as a circularly polarized electromagnetic wave).  
Setting aside Bell's theorem which forbids such an explanation and any other local realistic theory, will such a classic explanation ever be able to reproduce the probabilities obtained from QM? I have seen that deterministic QM interpretations like Bohmian mechanics introduce concepts like infinite dimensional configuration space. Why is this necessary for reproducing QM?
 A: There are several serious obstructions. One is provided by Bell's analysis concerning the conflict between realism and locality. However that obstruction concerns a very peculiar situation, referring to a bipartite system, with parts causally separated,  and quantum entangled states. 
There is another no-go result, usually called Kochen-Specker theorem, leading to a very severe obstruction against any completely classical interpretation of Quantum Mechanics based on hidden variables and epistemic randomness (however Bohmian quantum mechanics is untouched by it). 
Actually this theorem exists into a number of versions and its origin can be traced back to the celebrated Gleason's theorem as observed by Bell himself in his second famous paper of 1966 preceding the paper by Kochen and Specker of 1967.
The basic idea underpinning the no-go result is that quantum observables $A$ (selfadjoint operators on the Hilbert space of the system) are actually classical variables and there is a classical hidden state $\lambda$ (a set of hidden classical variables  $\lambda \in \Lambda$) which fixes the values $v_\lambda(A) \in \mathbb R$ of every observable $A$. 
In this view, randomness of values attained by  measurements of quantum observables is explained by assuming that $\lambda$  is unknown, but  we know only a probability distribution $\mu$ over $\Lambda$ describing the probability that $\lambda$ attains some value (discrete distribution) or stay in some "continuous" set. This is what happens, for instance, in classical statistical mechanics.  Here quantum probability becomes epistemic instead of ontic as in the standard interpretation of QM. In other words there must exist some correspondence $\mu \leftrightarrow  |\psi \rangle $ such that
$$\langle \psi| A \psi \rangle = \int_{\Lambda} v_\lambda(A) d\mu(\lambda)\:.$$
It remains to fix general rules to associate sharp values $v_\lambda(A)$ to observables $A$. The problem is how one should deal with functional relations as $C=A+B$. The naive idea to always assume that $v_\lambda(C) = v_\lambda(A) + v_\lambda(B)$ turns out to be untenable when $A$ and $B$ are quantistically described as incompatible observables  as explained by Bell analysing an earlier no-go theorem by von Neumann in 1966. 
A fair set of assumptions for $A \mapsto v_\lambda(A)$, which avoids to tackle any classical interpretation of quantum incompatibility, was proposed by Kochen and Specker referring to the algebra of observables $B(\cal H)_{sa}$ over a finite-dimensional Hilbert space $\cal H$ (finite dimensionality requirement can be relaxed by assuming some suitable continuity requirement on $v_\lambda$).
(1) The map $v_\lambda : B({\cal H})_{sa} \ni A \mapsto v_\lambda(A) \in \mathbb R $ is non-trivial (not all values are $0$).
(2) If $A,B \in B(\cal H)_{sa}$ are compatible observables (i.e. they commute), then
$v_{\lambda}(A+B) = v_\lambda(A)+ v_\lambda(B)$.
(2) If $A,B \in B(\cal H)_{sa}$ are compatible observables (i.e. they commute), then
$v_{\lambda}(AB) = v_\lambda(A)v_\lambda(B)$.
A more precise theory would also fix how the map $v_\lambda$ deals with incompatible observables. This thechnical specification is not necessary  for producing the no-go result I go to state and this fact also shows how KS' result is powerful. 
Kochen-Specker Theorem 
If $3\leq \dim(\cal H) < +\infty$, then  there is no map $v_\lambda : B(\cal H)_{sa} \ni A \mapsto v_\lambda(A) \in \mathbb R $ satisfying requirements  (1), (2), (3).
This theorem rules out from scratch every classical interpretation of QM where the realism hypothesis, i.e. every quantum observable is actually classical and always has an (unknown) sharp value. All that before any attempt to explain quantum randomness in terms of some classical uncertainty. 
Actually, a closer scrutiny shows that there is a way out when assuming the contextuality requirement: that the always existing values $v_\lambda(A)$ depend also on which observable $B$ I measure together with $A$ ($B$ is therefore assumed to be compatible with $A$). It may happens that $v_\lambda(A|B)\neq v_\lambda(A|B')$ if $B$ and $B'$ are incompatible (compatibility is not a transitive relation!). This impervious approach seems to be logically consistent even if it requires a big revision of our classical ideas on the physical world (personally I definitely prefer the standard interpretation of QM!).
The result of Kochen and Specker rules out realistic non-contextual hidden-variable interpretations of quantum theory. 
There is an equivalent formulation of K-S theorem which is more suitable for experiments. It is  based on the notion of test. A test is an observable which can assume only the value $0$ or $1$, in the standard formalism tests are all of orthogonal projectors $P \in B(\cal H)_{sa}$.
Kochen-Specker Theorem 
If $3\leq \dim(\cal H) < +\infty$, then  there is a set $\cal P$ of tests such that there is no map $v_\lambda: {\cal P} \ni P \to \{0,1\}$ satisfying the following requirements
(1) If $P,P' \in \cal P$ are compatible mutually exclusive tests ($PP'=0$ as orthogonal prjectors), then at most one of $v_\lambda(P)$, $v_\lambda(P')$ does not vanish.
(2)  If $P_1,\ldots, P_n \in \cal P$ is a set of pairwise compatible and mutually exclusive tests such that $P_1+\ldots + P_n =I$, then one of $v_\lambda(P_k)$ does not vanish.
The original proof of KS theorem in 1967 proved that if $\dim(\cal H)=3$ there is a set of 117 tests satisfying the theorem. Actually a general proof valid for every dimension (also infinite when assuming some continuity hypothesis on $v_\lambda$) easily arises from Gleason's theorem as already noticed by Bell.

I have seen that deterministic QM interpretations like Bohmian mechanics introduce concepts like infinite dimensional configuration space.

I do not think so. Bohmian mechanics for particles is formulated in the standard $3N$ dimensional configuration space of a system of $N$ particle. Maybe you are considering the system of a quantum field. I am not an expert on this subject however. 
As recent references I would like to mention various entries of Stanford Encyclopedia of Philosophy,  Landsman's book on foundations of quantum theory, a book consisting of a wide collections of recent papers on Bell's analysis and further foundational issues.
(I am publishing a book on fundamental mathematical structures in quantum theoryand chapter 5 is completely devoted to study these issues including Bell's inequality and its interplay with locality and contextuality).
My answer here could be of interest
A: 
Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics and the world as described by classical mechanics, particularly concerning quantum entanglement where two or more particles in a quantum state continue to be mutually dependent, even at large physical separations.

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Bell's theorem states that any physical theory that incorporates local realism cannot reproduce all the predictions of quantum mechanical theory. Because numerous experiments agree with the predictions of quantum mechanical theory, and show differences between correlations that could not be explained by local hidden variables, the experimental results have been taken by many as refuting the concept of local realism as an explanation of the physical phenomena under test. For a hidden variable theory, if Bell's conditions are correct, the results that agree with quantum mechanical theory appear to indicate superluminal (faster-than-light) effects, in contradiction to the principle of locality. 

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(Currently accepted quantum field theories are local in the terminology of the Lagrangian formalism and axiomatic approach.)

The "no go" means that all the data fitted with quantum mechanical models and thus validate quantum mechanics, cannot be fitted with classical theories if locality is assumed in the mathematical model. 
Locality is a principle in both classical and quantum physics, principles are axioms for physics models.

n physics, the principle of locality states that an object is directly influenced only by its immediate surroundings. A theory which includes the principle of locality is said to be a "local theory". This is an alternative to the older concept of instantaneous "action at a distance". Locality evolved out of the field theories of classical physics. The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence. 

So the answer is no, you cannot ignore Bell's theorem  within main stream physics which is what this site discusses.
