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For rational numbers $c$, we present a trichotomy of the set of totally real(totally $p$-adic, respectively) preperiodic points for maps in the quadraticunicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadraticpolynomials $f_c$ with rational parameters $c\in\mathbb{Q}$ so that $f_c$ hasonly finitely many totally real (totally $p$-adic, respectively) preperiodicpoints. These results rely on an adelic Fekete-type theorem and dynamics of thefilled Julia set of $f_c$. Moreover, using a numerical criterion introduced in[NP], we make explicit calculations of the set of totally real$f_c$-preperiodic points when $c=-1,0,\frac{1}{5}$ and $\frac{1}{4}.$

Preperiodic points with local rationality conditions in the quadratic unicritical family